a catalog of the real numbers - carnegie mellon...

A Catalog of the Real Numbers

by Michael Ian Shamos, Ph.D., J.D.

School of Computer Science Carnegie Mellon University

2007

Copyright © 2007 Michael Ian Shamos. All rights reserved.

Preface

Have you ever looked at the first few digits the decimal expansion of a number and tried to recognize which number it was? You probably do it all the time without realizing it. When you see 3.14159... or 2.7182818... you have no trouble spotting and e, respectively. But are

you on familiar terms with 0.572467033... or 0.405465108...? These are 2 3

12

and

, both of which have interesting stories to tell. You will find them, along with those of more than 10,000 other numbers, listed in this book. But why would someone bother to compile a listing of such facts, which may seem on the surface to be a compendium of numerical trivia?

log log3 2

Unless you had spent some time with 2 3

12

, you might not realize that it is the sum of two

very different series, ( )cos

12

1

k

k

k

k and . The proof that each sum

equals

1

)12()2(k

kkk

2 3

12

is elementary; what connection there may be between the two series remains

elusive. The number log turns up as the sum of several series and the value of certain

definite integrals:

log3 2

1

31k

k k

, ( )

1

2

1

1

k

kk k

,

)2)(1 xx

dx

1 ( and

0 2 xe

dx

1 . If your research led to

either number, this information might well lead you to additional findings or help you explain what you had already discovered.

Mathematics is an adventure, and a very personal one. Its practitioners are all exploring the same territory, but without a roadmap they all follow different paths. Unfortunately, the mathematical literature dwells primarily on final results, the destinations, if you will, rather than the paths the explorers took in arriving at them. We are presented with an orderly sequence of definitions, lemmas, theorems and corollaries, but are often left mystified as to how they were discovered in the first place.

The real empirical process of mathematics, that is, the laborious trek through false conjectures and blind alleys and taking wrong turns, is a messy business, ill-suited to the crisp format of referred journals. Possibly the unattractiveness of the topic accounts for the scarcity of writing on the subject. How many unsuccessful models of computation did Turing invent before he hit on the Turing machine? How did he even generate the models? How did he realize that the Halting Problem was interesting to consider, let alone imagine that it was unsolvable? What did his scrap paper look like?

This book is a field guide to the real numbers, similar in many ways to a naturalist’s handbook. When a birdwatcher spots what he thinks may be a rare species, he notes some of its characteristics, such as color or shape of beak, and looks it up in a field guide to verify his identification. Likewise, when you see part of a number (its initial decimal digits), you should be able to look it up here and find out more about it. The book is just a lexicographic list of 10,000 numbers arranged by initial digits of their decimal expansions, along with expressions whose values share the same initial digits.

PREFACE iv

This book was inspired by Neil Sloane’s A Handbook of Integer Sequences. That work, first published in 1971, is an elaborately-compiled list of the initial terms of over 2000 integer sequences arranged in lexicographic order. When one is confronted with the first few term of an unfamiliar sequence, such as {1, 2, 5, 16, 65, 326, 1957,...}, a glance at Sloane reveals that this sequence, number 589, is the total number of permutations of all subsets of n objects, that

is,

n

k

n

k

n

k jn

kn

n

k

nk

111 !

1!

)!(

!! , which is the closest integer to n . Sloane’s book is

extremely useful for finding relationships among combinatorial problems. It leads to discoveries, which help in turn generate conjectures that hopefully lead to theorems.

e!

What impelled me to compile this catalog was an investigation into Riemann’s -function that I began in 1990. It struck me as incredible that we have landed men on the moon but still

have not found a closed-form expression for ( )31

31

kk

. I say incredible because in the 18th

century Euler proved that ( )sk s

k

1

1

is a rational multiple of s for all even s, and gave a

formula for the coefficient in terms of Bernoulli numbers. No such formula is known for any odd s. It is not even known whether a closed form exists.

I began exploring such sums as 1

2 k ks sk t

, 1

1 k ks sk t

and ( )

1

2

k

s sk k k t

, whose values

can often be written as simple expressions involving various values of the -function. For

example, 1 5

43

5 32 k kk

( ), 1

6

1

24 21

2

k kk

coth and the remarkable

1 1

124 22 k kk

. To keep track of these many results, I kept them in a list by numerical

value, conveniently obtained using Mathematica®. Very quickly, after experimenting with

sums of the form , I began to notice connections that led to the following

theorem.

( )ak bk

11

Theorem 1. For an integer, b an integer such that a ba 1 2 and c a positive real

number, then 1

1 1ck kbj

aj b

ca bk

j

( )

For example, when I observed that the sums

.

An proof is given at the end of the chapter. The proof is elementary, but I never would have been motivated to write down the theorem in the first place without studying the emerging list of expressions sorted by numerical value.

After a time, I became somewhat obsessed with expanding the list of values and expressions and found that as it grew it became more and more useful. I began to add material not directly related to my research and found that it raised more questions than it answered. It became for me both a handbook and a research manifesto.

1

221

k

k

and ( )2

4 11

kk

k

of the form

shared the same 20-

digit prefix, this evoked considerable interest, since sums1

0 abk

k

are very difficult

PREFACE v

to treat analytically. After verifying that the sums w ual to ov

look for generalizations, exploring such sums as

ere eq er 500 digits, I began to 1

230

k

k ,

1

440

k

k ,

( )2

1

k

21k

k and

( )3

4 11

kk

k

give a few examples. The results suggested a conjecture, which stheorem:

, to

oon yielded the following

Theorem 2. For c > 1 real and p prime, 1

11 1c cpi

kp

kp

k

i

(, where

) ( )n is the

unction.

For a proof, see the end of the chapter. Alternatively, the theorem may be expressed in the

form

Moebius f

1

1

1

)(

kka

kp

1 /1kpp

k

a . This allows us to relate such strange expressions as

12

1k and

k 3

1k to Moebius function sums. Unfortunately, the theorem seems to

1 3k e

shed no light on trickier sums like 12

1 kk

k

and 1

1 k kk

k .

This b what is kna small space a potentially huge number of objects. It is similar to the drawers in the card catalog of a library. Imagine that a library only has 26 card drawers and that it places catalog

them. nd a b k by Taylor, you must go to drawer “T” and

ook is own in computer science as a hash table, a structure for indexing in

cards in each draw without sorting To fi oo look through all of the cards. If the library has only a hundred books and the authors’

names are reasonably distributed over the alphabet, you can expect to have to look at only two or three cards to find the book you want (if the library has it). If the library has a million books, this method would be impractical because you would have to leaf through more than 20,000 cards on the average. That’s because about 40,000 cards will “hash” to the same drawer. However, if the library had more card drawers, let’s say 17,576 of them labeled “AAA” through “ZZZ,” you would have a much easier time since each drawer would only have about 60 cards and you would expect to leaf through 30 of them to find a book the library holds (all 60 to verify that the book is not there).

Hashing was originally used to create small lookup tables for lengthy data elements. The term “hashing” in this context means cutting up into small pieces. Suppose that a company has 300 employees and used social security numbers (which have nine digits) to identify them. A table large enough to accommodate all possible social security numbers would need 109 entries, too large even to store on most computer hard disks. Such a table seems wasteful anyway, since only a few hundred locations are going to be occupied. One might try using just the first three digits of the number, which reduces the space required to just a thousand locations. A drawback is that the records for several employees would have to be stored at that location (if their social security numbers began with the same three digits). An attempt to store a record in a location where one already exists is called a “collision” and must be resolved using other methods. There will not be many collisions in a sparse table unless the numbers tend to cluster strongly (as the initial digits of social security numbers do). Using this scheme in a small community where workers have lived all their lives might result in everyone hashing

PREFACE vi

to the same location because of the geographical scheme used to assign social security numbers. A better method would be to use the three least-significant digits or to construct a “hash function,” one that is designed to spread the records evenly over the storage locations. A suitable hash function for social security numbers might be to square them and use the middle three digits. (I haven’t tried this.)

Getting back to the library, notice that any two cards in a drawer may be identical (for multiple copies of the same book) or different. The only thing they are guaranteed to have in common is the first three letters of the author’s last name. Now imagine that the library sets up 2620 drawers. Aside from the fact that you might have to travel for some time to get to a dra

authors have the same name, even if their nam

wer, once you arrive there you will probably find very few cards in it, and the chances are excellent that if there is more than one card, you are dealing with the same author or two authors who have identical names. Of course, it’s still possible for two authors to have different names that share the same 20 first letters.

This book is a hash table for the real numbers. It has 1020 drawers, but to save paper only the nonempty drawers are shown. But it differs very significantly from the card catalog of a library, in which the indexing terms (the “keys”) have finite length. When you examine two catalog cards, you can tell immediately whether the

es are very long. When you look up an entry in this book, you can’t tell immediately whether the expressions given are identically equal or not. That’s because their decimal expansions are of infinite length and all you know is that they share the same first 20 digits. With that warning, I can tell you there is no case in this book known to the author in which two listed quantities have the same initial 20 digits but are known not to be equal. There are many cases, however, in which the author has no idea whether two expressions for the same entry are equal, but therein lies the opportunity for much research by author and reader alike.

As a final example, seeing with some surprise that sin sink

k

k

kk k

1

2

21

1

2

led me to an

investigation of the conditions under which the sum of an infinite series equals the sum of the squares of its individual terms. (We may call such series “element-squaring.”) Element-squaring series rarely occur naturally but are in fact highly abundant:

Theorem 3. Every convergent series of reals B bkk

1

such that B bkk

( )2 2

1

can

be transformed into an element-squaring real series

B bk ing to B a single real by prependk 0

term b0 iff B B( )2 1 . Proof: see the end of the cha

I hope that these examples show that the list offers both information and challenges to the reader. It is a reference work, but also a storehouse of questions. Anyone who is fascinated by mathematics will be able to scan through the pages here entrie for a ber are related.

4

and wonder if, how and why the s given num

ts. The reader is encouraged to develop his own sup

pter.

The selection of entries in this book necessarily reflect the interests and prejudices of the author. This will explain the relatively large number of expressions involving the -function. For this I make no apology; there are more interesting real numbers than would fit in any book, even listing only their first 20 digi

plements to this catalog by using Mathematica to develop a list of numbers useful in his or her own field of study.

PREFACE vii

Professor George Hardy once related the now-famous story of his visit to an ailing Ramanujan in which Ramanujan asked him the number of the taxicab that brought him. Hardy thought the number, 1729, was a dull one, but Ramanujan instantly answered that it was the smallest integer that can be written as the sum of two cubes in two different ways, as 1 123 3 and

integer such that

9 103 3 . Hardy marveled that Ramanujan seemed to be on intimate terms with the integers and regarded them as his personal friends1. It is hoped that this book will help make the real numbers the reader’s personal friends.

Proofs of Theorems 1-3

Theorem 1. For a 1 an integer, b an a b 2 and c a positive real

num( )b

c k

aj

ckjj

a

1 1

1at t the chapter.

ber, then kb b . A proof is given he end of

Proof: The left-hand sum may be written as the double sum 1

11aj b j

kj

k c

order of summation,

. Reversing the

1111

11 11 1 k jk jj k

1

1jabjajbjbaj

ckkckkck , which equals

1 1

1

1

1 1k ck ck kbk

a a b bk

p

. QED.

Theorem 2. For c > 1 real and prime, 1

11 1c

kp

cpi

kpk

i

( ), where ( )n is the Moebius

function.

Proof: ( ) ( )kp kp

kp jkp

. Note thc ck jk 11 11

at every term of 1

i

1 c pi

sum, certainly in the case when

appears at least once in

the ouble 1 and j pik d . We will show that the sum of all

coefficients of 1

c pi in the double sum is zero except when k 1 is a power of p, in which and j

case ( )kp 1, and the result follows. Suppose that jk is a power of p. Then k ( )kpif 1, 0 since kp has a repeated factor.

Therefore, only the terms with k 1 contribute to the sum and ( )kp 1 each j a power of p. Now consider all pairs j, k with jk not a power of p. Suppose that i has the factorization i

p a1 th each of the a 0. pq , wi i If aq1 p k or if k has any repeated prime factors, then

( )kp 0. Since k i, it suffices to consider only those k whose factors are products of subsets P P o set f distinct primes taken from thek ppp q 1 , a set having cardinality q 1 or q,

on whether depending P contai If ns p. Pk as an odd number of elements, then h ( )kp 1; , if otherwise Pk has an even number of elements, then ( )kp 1. ( )kp 0 cannot be zero

since p does not divide k. But the number of subsets of Pk having an odd num

ber of elements

1This incident is partially related in Kanigel, R. The Man Who Knew Infinity. New York: Washington Square Press (1991). ISBN 0-671-75061-5. The book is a fascinating biography of Srinivasa Ramanujan.

PREFACE viii

is ing an even numbeble sum

the same as the number of subsets hav r of elements. Therefore, the terms cancel each other and terms with jk not a power of p do not contribute to the dou . QED.

Theorem 3. Every convergent series of reals B bkk

1

such that B bkk

( )2 2

1

can

be transformed into an element-squaring series

B bkk 0

by appending to B a single real term

b0 iff B B( )2 1

4 .

Proof: For B to be element-squaring, it is necessary that

b b b b B b Bkk

kk

2 2 2 2 2

( ) . Solving this quadratic equation for b yields 00

01

0

0

bB B

0

21 )1 4

( )(

2, which is real iff B B( )2 1

4 0 . Since is finite, b B b B0

that if

converges. It is also eleme y to showntar B B( )2 1

4 , then there is no real ele ent that m

can be prepended to B to m ke it element-squaring. QED.

a

Note that B does not necessarily imply that B( )2 b. For example, let kk

k

( )1

.

Then

2

1,

1B , which is finite, but 2

Bkk

( )2 1

1

, which ( )2Bdiverges. Likewise, does

not imply that . If bkk 1

, then B( )2

6

2 but B diverges. B

Using This Book

This book is a catalog of over 10,000 real numbers. A typical entry looks like:

.20273255405408219099... log log3 2

2

1

5

arctanh AS 4.5.64, J941, K148

= 1

5 2 1

1

22 10

1

11

kk

k

kkk k

( )

( )

=

12)12(

log

x

dxx

= dxx

xxx1

0

5sinsin)( GR 1.513.7

Each entry begins with a real number at the left, with its integer and fractional parts separated. Entries are aligned on the decimal point. Three dots are used to indicate either (1) a non-terminating expansion, as above, or (2) a rational number whose period is greater than 18. Three dots are always followed by the symbol , which denotes “approximately equal to.”

The center portion of an entry contains a list of expressions whose numerical values match the portion of the decimal expansion shown. The entries in the list are separated by the equal sign =; however, it is not to be inferred that the entries are identically equal, merely that their decimal expansions share the same prefix to the precision shown.

There is no precise ordering to the expressions in an entry. In general, closed forms are given first, followed by sums, products and then integrals. At the right margin next to expression may appear one or more citations to references relating the expression to the numerical value given or another expression in the entry. When more than one expression appears on a line containing a citation, it means that the citation refers to at least one expression on that line. See also, “Citations,” below.

Format of Numbers

Numbers are given to 19 decimal places in most cases. Where fewer than 15 digits are given, no more are known to the author. Rational numbers are specified by repeating the periodic portion two or more times, with the digits of the last repetition underlined. For example, 1/22 would be written as .04504545, which is an abbreviation for the non-terminating decimal .0454545454545045... . Since rational numbers can be specified exactly, the = symbol rather than is used to the right of the decimal. Where space does not permit a repetition of the periodic part, it is given once only. If the period is longer than 18 decimals, the sign is used to denote that the decimal has not been written exactly. No negative entries appear.

Ordering of Entries

Entries are in lexicographic order by decimal part. Entries having the same fractional part but different integer parts are listed in increasing magnitude by integer part. Rational numbers are treated as if their decimal expansions were fully elaborated. For example, these entries would appear in the following order:

USING THIS BOOK x

.505002034457717739... .5050150501 = .505015060150701508... .5050 = .505069928817260012...

Arrangement of Expressions

In general, terms in a multiplicative expression are listed in decreasing order of magnitude except that constants, including such factors as ( )1 k appear at the left. For example, 4 2 3k kk! is used in preference to 4 2 . Additive expressions are ordered to avoid initial minus signs, where possible. For example, 3

3k kk !2 k is used instead of 2 3k . These principles are

applied separately in the numerators and denominators of fractions.

Citations Abbreviations flushed against the right margin next to an expression denote a reference to

an equation, page or section of a reference work. Page numbers are prefixed by the letter p, as in “p. 434.” A list of cited works appears at the end of the book.

Acknowledgments

No book like this one has previously appeared, probably because of the monumental difficulty of performing the required calculations even with computer assistance. However, with the development of the Mathematica® program, a product of Wolfram Research, Inc., it has been possible to obtain 20-digit expansions of most of the entries without undue difficulty. The symbolic summation capability of that program has been invaluable in developing closed-form expressions for many of the sums and products listed. Even though my involvement with computers extends back almost 35 years, I have no hesitation in declaring that Mathematica is the most important piece of mathematical software that has ever been written and my debt to its authors is, paradoxically, incalculable.

The tendency to be fascinated with numbers, as opposed to other, abstract structures of mathematics, is not uncommon among mathematicians, though to be afflicted to the same degree as this author may be unusual. If so, it because of the role played by a collection of people and books in my early life. My grandfather, Max Shamos, was a surveyor who, to the end of his 93-year life, performed elaborate calculations manually using tables of seven-place logarithms. The work never bored him — he took an infectious pride in obtaining precise results. When calculators were developed (and, later, computers), he never used them, not out of stubbornness or a refusal to embrace new technology, but because, like a cabinetmaker, he needed to use his hands.

I read later in Courant and Robbins’ What is Mathematics, of the accomplishments of the calculating prodigies like Johann Dase and of William Shanks, who devoted decades to a manual determination of to 707 decimal places (526 of them correct). At the time, I couldn’t imagine why someone would do that, but I later understood that he did it because he just had to know what lay out there. I expect that people will ask the same question about my compiling this book, and the answer is the same. The challenge set by Shanks is still alive and well — every year brings a story of a new calculation — is now known to several billion digits.

My father, Morris Shamos, is an experimental physicist with a healthy knowledge of mathematics. When I was in high school, I was puzzled by the formula for the moment of

inertia of a sphere, M m r2 . That is was proportional to the square of the radius made

sense enough, but I couldn’t understand the factor of 2/5. Dad set me straight on the road to advanced calculus by showing me how to calculate the moment by evaluating a triple integral.

2

5

As a teenager during the 1960s, I worked for Jacob T. (“Jake,” later “Jack”) Schwartz at the New York University Computer Center, programming the IBM 7094. (That machine, which cost millions, had a main memory equivalent to about 125,000 bytes, about one-tenth of the capacity of a single floppy disk that today can be purchased for a dollar.) NYU had a large filing cabinet with punched-cards containing FORTRAN subroutines for calculating the values of numerous functions, such as the double-precision (in the parlance of the day) arctangent. By poring over these cards, I learned something of the trouble even a computer must go through in its calculations of mathematical functions. It was this exposure that drew me to field that later came to be called computer science.

ACKNOWLEDGMENTS xii

For the past 20 years I have been an enthusiastic user of Neil Sloane’s A Handbook of Integer Sequences in combinatorial research, so it is particularly pleasing to note that this book itself consists entirely of such sequences.

My greatest debt, however, is to Stephen Wolfram for his creation of Mathematica, the single greatest piece of software ever written. In developing this catalog, I used Mathematica on and off over a 15-year period to explore symbolic sums and integrals and to compute most of the constants that appear in this book to 20 decimal places.

Notation

a n( ) greatest odd divisor of integer n.

Ai x( ) Airy function, solution to y xy 0. AS 10.4.1

b x( ) ( )

1

0

k

k x k.

Bk kth Bernoulli number, B Bk k ( )0 . AS 23.1.2

B xk ( ) kth Bernoulli polynomial, defined by te

eB x

t

k

xt

t k

k

k

1 0

( )!. AS 23.1.1

bp n( ) number of divisors of n that are primes or powers of primes.

ck coefficient of xk in the expansion of ( )x .

C x( )

0

2

142

0

2

)14(2)!2(

)1(cos

2

1)(

kk

kkkx

kk

xdttxC

. AS 7.3.1, 7.3.11

Ci x( ) cosine integral,

x

k

kk

kk

xxdt

t

txxCi

0 1

2

)2()!2(

)1(log

1coslog)( .

AS 5.2.2, AS 5.2.16 d n( ) number of divisors of n, d n n( ) ( ) 0

dn kth derangement number, d d d d1 2 3 40 1 2 9 , , , , nearest integer to k e!/ . Riordan 3.5

E x( ) complete elliptic integral

dxxE 2/

0

2sin1)( .

En nth Euler number,

2

12 k

kk EE . AS 23.1.2

E xn( ) nth Euler number, defined by 2

1 0

e

eE x

t

k

xt

t k

k

k

( )!

. AS 23.1.1

Ei x( ) exponential integral

1 !

log)(k

k

x

t

kk

xx

t

dtexEi . AS 5.5.1

Erf x( ) error function

0

12

0

12

0 !)!12(

2)1(2

)12(!

)1(22 22

k

kkkx

k

kkxt

k

xe

kk

xdte

. AS 7.1.5

Erfc x( ) complimentary error function = 1 Erf x( ). AS 7.1.2 f n( ) number of representations of n as an ordered product of factors

Fn nth Fibonacci number, F F F F Fn n n1 2 11 2 ;

0 1F (; ; )a x hypergeometric function x

k a

k

kk ! ( )

0

, where a a a a kk( ) ( )...( ) 1 . Wolfram A.3.9

1 1F ( ; ; )a b x Kummer confluent hypergeometric function x a

k b

kk

kk

( )

( )!

0 ,

NOTATION 599

where a a a a kk( ) ( )...( ) 1 . Wolfram A.3.9

2 1F ( ; ; ; )a b c x hypergeometric function x a b

k c

kk k

kk

( ) ( )

( )!

0

, where a a a a kk( ) ( )...( ) 1 . Wolfram A.3.9

Catalan’s constant ( )

( ). ...

1

2 10 915965594177219015052

0

k

k k G

Gn

1 )13(

1

)13(

11

knn kk

gn

1 )13(

1

)13(

11

knn kk

gd z( ) Gudermannian, gd z ez( ) arctan 22

Hn harmonic number: Hkn

k

n

1

1

H ne even harmonic sum: H

ke

n

k

n

1

21

H no odd harmonic sum: H

ko

n

k

n

1

2 11

j( )n generalized harmonic number: H

kj

n

jjj

n jk

n j j( )

( )

( )( ) ( )

( )!,

1 1 1

11

1

1 1

H

hn

0 )56(

)1(

)16(

)1(

kn

k

n

k

kk

hg x( ) Ramanujan’s generalization of the harmonic numbers, hg xx

k k xk

( )( )

1

I xn( ) modified Bessel function of the first kind, which satisfies the differential equation

z y zy z n y2 2 2 0 ( ) . Wolfram A.3.9

jn

1 )13(

1

)13(

11

knn kk

J311

J xn( ) Bessel function of the first kind ( )

!( )!

1

2

2

20

k n k

n kk

x

k n k LY 6.579

k 1 2 k

K x( ) complete elliptic integral of the the first kind

dx

2/

02sin1

1

K xn( ) modified Bessel function of the second kind, which satisfies the differential equation

z y zy z n y2 2 2 0 ( ) . Wolfram A.3.9

l x( )

n

kk k

k

xk

xk

k

k

11

log)log(log , when x is an integer n.

NOTATION 600

li x( ) logarithmic integral x

t

dtxli

0 log)(

Li xn( ) polylog function Li xx

kn

k

nk

( )

1

log x natural logarithm loge x

a prime number. denotes a sum over all primes. pp

pd n( ) product of divisors pd n dd n

( )|

pf n( ) partial factorial pf nkk

n

( )!

1

0

pfac n( ) product of primes not exceeding n, pfac n pp n

( )

pq n( ) product of quadratfrei divisors of n, pq n pr n( ) ( ) 2

pr n( ) number of prime factors of n

r n( ) 4 ( )|

dd n HW 17.9

s n( ) smallest prime factor of n

sd n( ) sum of the divisors of n, sd n n( ) ( ) 1

S x( ) 34

1

2

0

2

)34()!12(

)2/()1(

2sin)(

k

k

kkx

xkk

dtt

xS

. AS7.3.2, AS 7.3.13

S n m1( , ) Stirling number of the first kind, defined by the recurrence relation:

x x x n S n m xm

m

n

( )...( ) ( , ) 1 1 1

0

. AS 24.1.3

S n m( , )2 Stirling number of the first kind, nm

k

km kk

m

mmnS

02 )1(

!

1),( . AS 24.1.4

Si x( ) sine integral,

x

k

kk

kk

xdt

t

txSi

0 0

12

)12()!12(

)1(sin)( . AS 5.2.1, AS 5.2.14

T s( ) 1

ksk , where k has an odd number of prime factors

v1

2

23)31(

2

23)31(1

ii

iiv

w x( ) w x e erfc ixix

kx

k

k

( ) ( )( )

( / )

2

2 10

Wn

0 )56(

)1(

)16(

)1(

kn

k

n

k

kk

z x2 ( ) z x k xx

x xk

k2

2

12

11( ) ( ( ) ) ( ( ))

NOTATION 601

( )s ( )( )

( )s

k

k

sk

1

2 1

1

0

Euler's constant

n k

n

knlim log

1

1

0.57721566490153286 AS 6.1.3

f

n

kn

dxxfkf1 1

)()(lim

( )s Riemann zeta function 1

1 ksk

( )m ( )s derivative of the Riemann zeta function ( log )

k

k

m

sk 1

( )s ( )( )

( )sk

sk

s( )

k

s

1

1 21

1

1

( )s ( )( )

( ) (sk

ss

)k

s

1

2 11 2

0

( )n Moebius function, ( ) , 1 1 ( ) ( )n k 1 if n has exactly k distinct prime factors, 0 otherwise. ( )n number of different prime factors of n.

k n( ) sum of kth powers of divisors of n, dk

d n|

golden ratio, 1 5

21 61803397498948482

. ...

( )n Euler totient function, the number of positive interegs less than and relatively prime to n.

( )n polygamma function ( )nkk

n

1

1

1

( )x kth derivative of the polygamma function ( ) ( )( )k

k

kx

d x

dx

( )n ( ) ( )n kk

n

1

( )s )()1(

21)( 2/ ss

ss s

( )n overcounting function ( )n -1 + the number of divisors of the g.c.d of the exponents of the prime factorization of n.

.00000000000000000000 = ( ) ( )

!

( )

!

1 1 1

0

1 2

1

k

k

k

k

k

k

k

k GR 1.212

= ( ) ( )

!

( )

!!( )

1 2 2 11

2

20 1

2

1

k k

k

k

k

kk

k

k

k

k

k

k

k

k

= ( )k

kk

1

Berndt Ch. 24 (14.3)

= ( )k

kk 2 11

= ( )sin

1 13

1

k

k

k

k

= log2

20 1

x

x

1 .000000000000000000000 = 1

11 k kk ( )

J396

= 1

2 21 kk

½

= k

kk ( ) !

11

GR 0.245.4

= 2

20

k

k

k

k( ) !

= 1

10 ( )!k k!k

= 2 1 3 3 1

3 3

4 3

72

2

6 5 41

k k

k k

k k

k k k k3k k

= 2 1

4 1

2 1

2 11 1

k

k k

k

k kkk k

( )

( )!

( )! ( )

!

=

( ) logk k

kk 1

= F k

kk

4 2

1 8

=

22 22k

k

kk

H

= ( )kk

12

=

2

)(

k k

k

=

( ) logk k

kk 1

Berndt Ch. 24, Eq. 14.3

= 1

1

S

, where S is the set of all non-trivial integer powers

(Goldbach, 1729) JAMA 134, 129-140(1988)

=1

21 n

k

k

n

kn !

log

!

= ( )sin

121

1

k

k

k

k

= 11 1

1

( )k

k k J1065

= dx

x

x dx

x21

21

log

= dx

x( ) /2 30 1

2

= dx

x x( )1 1 20

1

= x x

edxx

log( )1

0

=

cossin

xx

x

dx

x0

Berndt I, p. 318

= arctan x

xdx2

1

2 .000000000000000000000 = (½, , )!

( )! ( )!

!

1 02

2

2 11 1 1

k k

k

k

kkk

k

k k

= ( )

( )

k H F

k

k

kk

kk

kk

kk

k

1

2 2

1

4 1

2

1 1 0

=

1

1)2(exp

k k

k

=

12

)(

k k

kz

= 11

2 11

2

22

k k

k

kk k( )

= 11

220

k

k

= e ek kk

k

k

k

( ) / /

1

1

1 2

1

1

= log

(cos )

2

21

2

20

2

1

x

xdx

x

edx

dx

o

= x x

xdx

sin

cos

/

10

2

GR 3.791.1

= logsin cos/

x x d2

0

2

x

= e dxx

0

1

=

0

11

3xe

dxx

3 .000000000000000000000 = k k

kkk 6

2 1

0

4 .000000000000000000000 = k k k

k

k

kkk

kk

kk2

1

2

1

4 1

2 21

1 2 0

( )

( )

= ( )!!

( )!! ( )

( )!!

( )!! (

2

2 1

2

2 122

121

k

k k k

k

k k kk k

)

6 .000000000000000000000 = (½, , )

2 02

2

1

kk

k

= k

k kk

2

3 2 2( )( )

= x dx

e

xdx

xx

3

0

3

21

log

10 .000000000000000000000 = ( )k F k

kk

kk

k

4

2 20 1

12 . 000000000000000000000 = ( )

log

k dx

x xxe dxk

k

x

1

2

2

0 2 0

15 . 000000000000000000000 = ( / , , )1 3 4 03

4

1

kk

k

20 . 000000000000000000000 = k

k kk

2

4 3 3( )( )

24 . 000000000000000000000 = log4

20

xdx

x

26 . 000000000000000000000 = (½, , )

3 02

3

1

kk

k

52 . 000000000000000000000 = ( )k

kk

1

2

3

0

70 . 000000000000000000000 = k

k kk

2

5 4 4( )( )

94 . 000000000000000000000 = F kk

kk

2

1 2

150 . 000000000000000000000 = (½, , )

4 02

4

1

kk

k

1082 . 000000000000000000000 = (½, , )

5 02

5

1

kk

k

1330 . 000000000000000000000 = F kk

kk

3

1 2

25102 .000000000000000000000 = F kk

kk

4

1 2

1 .00000027557319224027... 1

101 ( )k !k

... 697

6613488

8

8

W J313

...

648000, the number of radians in one second

1 .000006974709035616233... 2coth

... 10

1 .0000115515612717522... 547 61

2 3 5

1

6 5

1

6 1

7

10 9

1

7

1

71

( )

( )

( )

( )

k k

k k k J328

1 .000014085447167... W7 J313

.0000248015873015873... 13

1 ( )k

!k

... 9

... sec1

.00004439438838973293... 1

22528

11

121

23

12

1

4096 12 1 122

Fdx

x, , ,

... e10

1 .0000513451838437726... ( )

( )

( )9

511 9

512

1

2 1 90

kk

AS 23.2.20

1 .00005516079486946347... 1

358318080

1

12

5

12

7

12

11

125 5 5 5 ( ) ( ) ( ) ( )

= ( )

( )

( )

( )

1

6 5

1

6 1

1

6

1

61

k k

k k k

1 .0000733495124908981... W6

691

87480

J313

... 8

... e9

...

( ) ( )log3

3

21

2k

kk

1 .000155179025296119303... ( )

( )

( )8

17

161280

255 8

256

1

2 1

8

80

kk

AS 23.2.20

1 .0002024735580587153... 3

31

.0002480158734938207... 1

81 ( )k !k

...

1

8192

1

16

1

16 12

30

( )

( )kk

.0002499644008724365... 16 6 112 6

2 1k kk

k k

( )

1 .0002572430074464511... 5 61

2 3

1

6 5

1

6 1

7

7 6

1

5

1

51

( )

( )

( )

( )

k k

k k k J328

.00026041666666666666 = 1

3840

1

5 25

!

.0002908882086657216...

10800, the number of radians in one minute of arc.

.000291375291375291375 =1

34321

492

8

kk

...

( )

( )( )3

4096

1

81921

1

162

31

kk

... 7

... e8 1 .00038992014989... W5 J313

1 .000443474605655004... ( )

( )

( )

( )

1

4 3

1

4 1

1

7

1

71

k k

k k k J326

1 .0004642141285359797... 1

7776

1

6

1

6 13

41

( )kk

1 .00047154865237655476... ( )

( )

( )7

127 7

128

1

2 1 70

kk

AS 23.2.20

1 .00049418860411946456... ( )11111

1

kk

1 .00056102267483432351...

1

3456

1

12

1

12 112

31

( )

( )kk

1 .000588927172046... r( )16 Berndt 8.14.1

5

82

1 2

4 2

2 2

16

2 2 2

2 2 2

2 2

16

2 2 2

2 2 2log

log( )log log

... 11

22

21

24

9

23

15

25

1

2

1 3

51

( ) ( ) ( ) ( )( )

( )

k

k

k

k

.00069563478192106151... ( )

( )

3

1728

1

12 31

kk

... ( )( )

12

21

2

1

kk

k

k k

.00088110828277842903...

1

3456

11

12

1

12 12

31

( )

( )kk

.00091188196555451621... e7

1 .00097663219262887431... 15

1 k kk

1 .00099457512781808534... ( )10110

1

kk

.00099553002208977438... ( )10 11

kk

.00101009962219514182... 13

124 6 4 1

1

110 4

2

( ) ( )kk kk k

.00102614809103260947... ( )5 5 11

110 5

2

kk kk k

.001040161473295852296... 6

.0010598949016150109... 15

8 42 6 4 6 1

1

110 6

2

coth ( ) ( ) ( )k

k kk k

.00108225108225108225 =1

9241

362

7

kk

.00114484089113910728...

1

3456

5

6

1

12 22

31

( )

( )kk

1 .00115374373790910569... 1

124416

1

12

5

12

7

12

11

123 3 3 3 ( ) ( ) ( ) ( )

=( )

( )

( )

( )

1

6 5

1

6 1

1

4

1

41

k k

k k k

1 .001242978120195538...

1

1458

1

9

1

9 82

31

( )

( )kk

1 .00130144245443407196... 1

31457280

1

8

3

8

5

8

7

85 5 5 5 ( ) ( ) ( ) ( )

=( )

( )

( )

( )

1

4 3

1

4 1

1

6

1

61

k k

k k k

.001311794958417665... 197

27 64

3 3

32

1

2 8 30

( ) ( )

( )

k

k k

.00138916447355203054... e

e kk4

1

41

1

2

1

4 21

cos

( )!

3 .00139933814196238175... 2

1 .0013996605974732513... 3 3

62

3

4

3 1

312

log

log log( )( )r Bendt 8.14.1

.00142506011172369955... 1

102 k kk log

.00144498563598179575... ( )2

62

k

kk

.00144687406599163371... ( )2

162

k

kk

1 .0014470766409421219... 6

61960

663 6

64

1

2 1

( )( )

( )kk

AS 23.2.20

...

1

8192

9

16

1

16 92

30

( )

( )kk

... ( ) ( )log4

4

21

2

k

kk

.00153432674083843739...

1

3456

3

4

1

12 32

31

( )

( )kk

.0015383430770987471... 930 5

1920 2 1

6

61

( )

( )

k

kk

.0015383956660985045... 4 3

256

1

4 8 30

( ) ( )

( )

k

k k

1 .001712244255991878...

1

1024

1

8

1

8 72

31

( )

( )kk

1 .00171408596632857117... 5 3

6

( )

... 1

1536

1

4

1

4 13

40

( )

( )

kk

.001815222323088761... 5

54

197

216

1

1 2 3

2

2 2 21

2

k k k kk ( ) ( ) ( ) LY 6.21

.001871372759366027379... 7

360

3

2

1

1

3

3 21

( )

k e kk

Ramanujan

=

32

1 1

( )k

e kk

.001871809161652456210... 1

12 21

22

1k e

k

ekk

kk

( )

.001872682449768546116... 1

121

12

1k e

k

ekk

kk

( )

.00187443047777494092... 1

1 121

02

1e

k

ekk

kk

( )

.001912334879124743... 15 109

960

2

6

2

0

e

k

k

k

( )!

1 .001949804343... j9 J311 1 .00195748553... J309 G9

2 .001989185473323053... 1110

1

kk

1 .00200839282608221442... ( )9

.00201221758463940331... ( )9 11

119

2

kkk k

.00201403420628152655... 3

46

3

25

1

23

5

44

9

83

17

162

387

642 ( ) ( ) ( ) ( ) ( ) ( )

= H

kk

k

15

1 2( )

.00201605994409566126... ( )8 1 11

19

2

kk kk k

.00201681670359358... 3277

3375 32

1

2 7

3

30

( )

( )

k

k k

.00202379524407751486... ( )7 2 11

19 2

2

kk kk k

.002037712107418497211... )7()3()5()10(4

7)1,9( 2 MHS

.00203946556435117678... ( )6 3 11

19 3

2

kk kk k

... 7 3

4096

1

8192

1

2

1

16 82

30

( )

( )( )

kk

.00210716107047916356... ( )5 4 11

19 4

2

kk kk k

.0021368161356763007...

1

3456

2

3

1

12 42

31

( )

( )kk

.00213925209462515473... ( )4 5 11

19 5

2

kk kk k

1 .0021511423251279551... 5

486

4

4

W J313

... 19

11520

2

96

2

48

3 2

8

5 3 2 4

log log ( ) log

=1

4 2 1

25

0k

k k

k

k( )

.00227204396400239669...

1

1458

8

9

1

9 12

31

( )

( )kk

1 .00277832894710406108... cosh cos1 1

.00228942996522361328... 1

3 6 19 6

1 1k kk

k k

( ( ) )

.00234776738898358259... ( )

( )

3

512

1

8 31

kk

... log

logsin/2

4 2 0

4

G

x dx GR 4.224.2

=

log/ x

xdx

1 20

1 2

GR 2.241.6

1 .0024250560555062466... 2 2

5

5

4

3

2 5

1 5

210

log loglog ( )

r Berndt 8.14.1

.002450822732290039104...

dx

e

exx

x

2

4

5 14

3

.00245775047043566779... 171109

24 5

2

1

ek

kk

( )!

.002478752176666358... e6

2 .0025071947052832538... 6 2 8 2 62 2

22

0

log log( )

k H

kk

kk

.00260416161616161616 =1

284

1

4 24

!

... 1

2

1

2

1

2

1

4 4

1

2

1

2

1

4 160

cos cosh cos( )!

e

e

k kk

... 1

8

5

3

1

4

1

4 7 12

log arctan

dx

x x

... dx

x72 1

... loglog

263

6 72

dx

x x

... dx

x x7 22

... 1

4

5

3

1

8 7 32

log

dx

x x

... cosh cos1 1

... loglog

27

3

1

24 7 42

dx

x x

.00286715218149708931... 1

92 k kk log

...

1

3

1

)5(

)1(

4

)3(3

216000

195353

k

k

k

=

1

0

1

0

1

0

555

1dzdydx

xyz

zyx

5 .0029142333248880669... 2

10

k

k k !

.002928727553540092... ( ) ( )

( )

3

36 324

1

3

2 1

31

k

k

k

k

...

1

8192

7

16

1

16 72

30

( )

( )kk

.0031244926731183736...

1

3450

7

12

1

12 52

31

( )

( )kk

.00318531972162645303... 2

33

2 33 2

2ex

x x

sinsin

... loglog

23

2

9

24 7 52

dx

x x

.00326776364305338547... 5

...

1

1458

7

9

1

9 22

31

( )

( )kk

... Re ( ) i

.0033104481564221302... 1

16

7

45 256

4

.0033178346712119643... 3 3

32

7

64

1

2 6 30

( ) ( )

( )

k

k k

=

1

0

1

0

1

0222

555

1dzdydx

zyx

zyx

24 .00333297476905227... e e e e

e e

k

ekk

( )

( ), ,

3 2

5

4

1

11 11 1

1

14 0

... 15 5

16

31

32

1

3 50

( ) ( )

( )

k

k k

.00337787815787790335...

1

1024

7

8

1

8 12

31

( )

( )kk

.003472222222222222222 =1

288 1 2 3 4 51

k

k k k k kk ( )( )( )( )( )

.00364656750482608467... 1

27

3

36

1

3 3

1

31

( ) ( )

( )

k

K k

=

1

0

1

0

1

0333

555

1dzdydx

zyx

zyx

...

3

2

36 3

91 3

216

1

432

1

6

( ) ( ) Berndt 7.12.3

= 1

6 5 31 ( )kk

.0037145966378051377... log( log( log(log ))) 2 1 .00372150430231012927... log( log(log ))2

.00374187319732128820... coth

12

12e

1 .00374187319732128820... coth

e e

e e

=3

800

1

2 3 41

5

( )( )( )(k k k k )k

... 19

2 2

1

4 3

1

4 1

5

12

1

51

1

5

( )

( )

( )

( )

k

k

k

k k J 326

1 .003795666917135695...

4 3 2 5

8 2 1 40

2arctan log dx

x

.00388173171751883... 32

256

1

4 2

3 1

31

( )

( )

k

k k

... 22

72

2

3

1

2 1

3

0

log( )

( )

k

kk

k

k

1 .0038848218538872141... arctan2

... j8 J 311

... 14

1 k kk

... 656

6200145

8

8

G J 309

.00396825396825396825 =1

2525 1

252

6

( )kk

.00402556575171262686... 7 4

8

7

8 4

18 4

2

( )

sinh

( )

k

k k k

=( ) ( ) ( )

int

1 1

142

4

1

k

k a non trivegerpower

k

k

ial

.004031441804149936148...

02

2

3 18

1dx

e

exx

x

.00406079887448485317... 19450 8 1

88

( )

( )

1 .0040773561979443394...

8

94508 ( )

.00409269829928628731... 15

16 8

2

8

2 2

2 2

cothsin sinh

cos cosh

= ( )8 11

118

2

kkk k

... 10

... 17 1 18

2 1k kk

k k

( )

... 3

42

2 3

3

2

( ) tanh

= 16 2 18 2

2 1k kk

k k

( )

.004147210828822741353... 1

12 52 ( )kk

... dxx

x

1

0

22

1

arccos

2421616

.004166666666666666666 =1

2407 ( )

.004170242045482641209... )6()3()5()4()7()2()9(4)1,8( MHS

... 15 3 18 3

2 1k kk

k k

( )

... 15

84

4 ( ) coth

= 14 4 18 4

2 14

1k kk

k

kk k k

( )( )

.00442764193431403234... ( )

( )( )

( )

2

2

7

1803 3 1

1

2

4 1

41

2

k

k

k

k

... 1

486

1

3

1

3 13

40

( )

( )

kk

.00450339052463230128... ( )2

52

k

kk

.00452050160996510208... ( )2

152

k

kk

...

2

2

22 2

2

82 2 1 2

cos sin csc

= k kk

kk

k k

2 2 12

21

2

2 22

( )( )

... 31 5

325

1

2 1 50

( )( )

( )

kk

AS 23.2.20

... 3

256

1

9

1

2 1 2 1 2 3

2

2 21

2

( ) ( ) (k k k )k

LY 6.21

... 12

3

64

1

2 3

3 1

31

G k

k

k

k

( )

( )

... log2661

960 7 62

dx

x x

... 13 5 18 5

2 1k kk

k k

( )

...

1

8192

3

8

1

16 62

30

( )

( )kk

... Gx

xdx

log log( )2

2

1

1

2

20

1

GR 4.295.6

=

log sin

/ xdx

20

2

... log log ( )22

41 2 8 r

.0047648219275670510... 67

64 96 360

3

8

5

2

1

2

2 4

52

( ) ( )

( )k kk

... 7

216

1

6 5

1

6 1

3 1

31

1

3

( )

( )

( )

( )

k

k

k

k k

... 7 3

1728

1

12 6 31

( )

( )

kk

.004915509943581366... sin

2

2 4

2

22

1

kk

.004983200... mil/sec

.00506504565493641652...

1

1458

2

3

1

9 32

31

( )

( )kk

.00507713440452621921... 4

51

93 5

192 2 1

( )

( )

k

kk

... 4

3

10 3

81

=kk

k

F Fk

2

12 1 2 12

1

22 2

3

2

1

4

1

33 3

5

2

1

4

, , , , , ,

...

1

3

1

)4(

)1(

1728

1549

4

)3(3

k

k

k

=

1

0

1

0

1

0

444

1dzdydx

xyz

zyx

.0051783527503297262... 7 3

128 512

1

1024

3

4

1

8 2

32

31

( )

( )( )

kk

.0053996374561862323... 15

42 4 6 2 6

1

( ) ( ) ( ) ( )kk

1

=1

8 62 k kk

=1

180 2 3 4 51

k

k k k kk ( )( )( )( )

6 .005561414313810887... 2

2 23

2

0log

x

dxx

... ( , )5 3

1 .00570213235367585932... 1

2

2

3

3

4

1

216 631

log log

k kk2 Ramanujan] Berndt Ch. 2

... 1

81 k kk log

.005899759143515937... 45 7

8 6

( )

Berndt 9.27.12

1 .005912144457743732... 1

250

1

5

1

5 42

31

( )

( )

kk

1 .0059138090647037913...

3

1 33

243

91 3

486

1

324

1

6

1

17496

1

6

( ) ( ) ( )

=( )

( )

2 1

6 1

2

40

k

kk

... 3

3032

26

27

1

2 5

( )

( )

k

k k

.00603351696087563779... log k

k 7

...

1

4096

7

162

2

8 81

( ) log x

x xdx

...

1

3375

7

152

2

8 71

( ) log x

x xdx

...

( ) log( )3

196

1

2744

1

22

2

8 61

x

x xdx

...

2 )(

)1(

)1(

)(

k k

k

k

k

... k

kk2

1 1

...

1

2197

7

132

2

8 51

( ) log x

x xdx

...

1

1728

7

122

2

8 41

( ) log x

x x

...

1

42

2

16

5 2 2

5 2 2arctan log

( )

arctan arctanlog1 2

8

2

81 2 1 2

3

8

2

16 6 22

dx

x x

31 .0062766802998201758... 3 2

0 1

log( )

xx

x xdx GR 4.261.10

= x e

edx

x

x

2 2

1

/

GR 3.4.11.16

.00628052611482317663... 2 1

3148

2

8

3 3

32

1

2 2

2

log ( ) ( )

( )

k

k

k

k

00629484324054357244... 1

160

5

61

11

6

1

64 12 1 62

Fdx

x, , ,

...

1

1331

7

112

2

8 31

( ) log x

x xdx

... log log21

531 6

2

dx

x x

= 4

6251

41

3

1

( )kk

k

k

...

1

1000

7

102

2

8 21

( ) log x

x x

.006476876667430481... log arctan3

4

2

2

1

2 4 6 22

dx

x x

... e e e

e e

k

ekk

2

4

3

0

4 1

1

13 0

( )

, ,

... 5

3

...

1

729

7

92

2

8 11

( ) log x

x xdx

... ( ) ( )i i

... log log

( )

2

4

1

6 1 50

1

x

xdx

...

( )2 1

2 11

kk

1

k

... e5

...

1

512

7

8 12

2

81

( ) log x

xdx

... ( )( )

( )4 1

12

422

24

1

jk

f k

kji k

.00677885613384857755... 3

46 5

1

23 4 3 22 ( ) ( ) ( ) ( ) ( ) ( ) 5

= H

kk

k

15

1 1( )

... log

arctan7

6

1

3

5

3

3

6

1

8 6 32

dx

x x

... 2

1

28

11

108

arctan

...

962 2 8 2 12 3

1

4 2 1

22 3

40

log log ( )( )

kk k

k

k

... 2 3

343

2

81

( ) log

x

x xdx

.00702160271746929417... 7

48 180

3 3

4

23

8

1

1 2

2 4

41

( )

( )( )k k kk

... 16

3

128 1

3 2

2 31

Gx

xdx

log

( )

... 9

... 3

8

1 1

4

1

1

e k

k

k

( )

( )!

1

... ( ) ( )

1 21 4

1

k

k

k

1 .007160669571457955175... 1

230 kk

... 1

2454 3 2 452 ( )

.0071918833558263656... i i ii i cos( log ) sin( log )

...

1

216

7

62

8 21

( ) dx

x x

2 ...

1

216

1

62

1

216

7

6 12 2

2

60

1

( ) ( ) log x dx

x

... sech2

1

( )kk

... 17

360

3 3

4

1

2

4 1

41

( ) ( )

( )

k

k

k

k

... Gk

k

k

10016

11025

1

2 7

1

21

( )

( )

... 1

6

1

22

1

csch ( )k

k

[Ramanujan] Berndt Ch. 26

= 1

1329 ( )

.00763947766738817903... log3

2

13

24 6 42

dx

x x

... j7 J 311

... 1

71 ( !)kk

... 1716 15428

15

2

135

1

12

4 6

7 71

k kk ( )

... 119

72 6

1

1 2

2

2 21

3

k k k kk ( ) ( ) ( )

1 .00788821... J 309 G7

...

1

125

7

52

2

8 31

( ) log x

x xdx

... 3 5

44

4 4

( )( ) c Berndt 9.27.12

.008062883608299872296...

dx

e

exx

x

2

2

3 14

1

.00808459936222125346... 4

1 22

2

0

2

log logx e x dxx

...

1

8192

5

16

1

16 52

30

( )

( )kk

2 .00815605449274531515... 11 1 1 14

1 8 3 8 5 8 7 8

sin( ( ) sin( ( ) sin( ( ) sin( ( )/ / / /

= 11

81

kk

... 1

614496 1536

1

4

3

43 3 3

G ( ) ( )

... ( )( )

3 13 3

32

f k

kk

Titchmarsh 1.2.14

...

1

3456

5

12

1

12 52

30

( )

( )kk

.00828014416155568957... ( )

( )

7 1

7

... 1

2

1

3

2

3

3

2 3

1

3 2

1

1

e

e

k

k

k

cos( )

( )

!

1 .00833608922584898177... sin sinh sin

( )

1 1

2

1

2 4

1

4

1

4 10 !

e

e kk

... 1

81 1

1

8 41

cos sin( )

e

k

kk 1 !

1 .00834927738192282684... ( )7

=

0

624

)42)(32)(22)(12(4

)2(

381

64

381

)5(64

1143

)3(8

kk kkkk

k

=

0

236

)72)(62)(52)(32)(12(4

)2()1175793942604248(

7560 kk kkkkk

kkkk

Cvijović 1997

... ( )8 1 11

17 1

2

kk kk k

.008650529099561105501... )11(2

331)9()2(84)7()4(21)6()5(4)4,7( MHS

... 4

... ( ) log( )6 1 1 1

1

6

2

k

k

k

kk k

... 1

4

2

3

1

3

1 3

2

1 3

2

i i

= 16 1 17

2 1k kk

k k

( )

.0085182264132904860...

1

1458

5

9

1

9 42

31

( )

( )kk

... 15 2 17 2

2 1k kk

k k

( )

.008650529099561105501... )5()3(7560

)1,7(8

MHS

.00866072400601531257...

1

1024

5

8

1

8 32

31

( )

( )kk

...

1

64

7

4

1

64

3

4

2

272 2

2

8 41

( ) ( ) log x

x xdx

... 7

83

2

1

4

( ) ( )i i

= 13 4 17 4

2 1k kk

k k

( )

1 .00891931476945941517...

( )

( )

( )6

7 71

k

kk

Titchmarsh 1.2.13

5 .00898008076228346631... 1

20 ( )!kk

.00898329102112942789... i i3

... 91 3

216 36 3

1

432

5

6

1

6 1

32

31

( )

( )( )

kk

.009197703611557574338... 3

8 4 3 3 3

4 1

91

cot coth

( )kk

k

...

02 )7)(4(!

2)1(

4

7

4

52

k

kk

kkke

... 3 3 23

1 12

21

2

31

( ) log ( )

( )

k

k

k

k

... 5

3

2

3 90

1

3

3 3

2

3 3

2

4

i i

= 13 4 17 4

2 1k kk

k k

( )

... 7

45 16

15

16

7 4

8

15

16

1

3

4

40

( ) ( )

( )

k

k k

2 .00956571464674566328...

161

4

643

41

22

40

1Gx dx

x

log

...

96 2 2

1

8 7 8 5 8 3 8 11

( )( )( )(k k k kk )

J243

.00960000000000000000 = 6

625

3

61

log x

xdx

... ( )

( )

3

125

1

5 31

kk

... 64

9

8

)3(

=

1

0

1

0

1

0222

555

1dzdydx

zyx

zyx

... 7

108 1

3 2

60

log x

xdx

... 5

2

... 9 3

961

2 2

21

31

( )( )

( )

k

k

k

k

.0099992027408679778... 1

3072

1

4

3

4 63 3

3

( ) ( )

4

... 2

33

1

6

1

320

csch

( )k

k k

... i i i i2

4 2 2 21 1 2 2 ( ) ( )

i

2

2

2 2coth

=H

kk

k2 1

21

... 3 3

2

1549

864

2

6 51

( ) log

x

x xdx

... 4

...

1

27

7

32

2

8 51

( ) log x

x xdx

... 69

16

3

8

3

2

1

1 2

2

2 31

( )

( ) ( )k k kk

... 7 3

16 64

1

128

1

4

32

( ) ( )

Berndt 7.12.3

=

1

1024

1

8

5

8

1

4 32 2

31

( ) ( )

( )kk

.01037718999605679651... ( )k

kLi

k kk k

1 16

26

2

1

... 11

2

1

2

1

4 4

1

1

cos cosh( )

( )!

k

kk k

GR 1.413.2

.01041666666666666666 = 1

96 1 2 3 4 5

2

1

k

k k k k kk ( )( )( )( )( )

1 .01044926723267323166... 41

41

1

16 2 21

sinh

kk

.01049303297362745807... 16

7

8

1

56

2

2

2

8 8

3

8cot

loglogsin logsin

=1

64 8

1

822 2k k

k

kk

k

( )

...

0

3)4(

)1(

4

)3(3

216

197

k

k

k

=

1

0

1

0

1

0

333

1dzdydx

xyz

zyx

... 1

24576

1

8

3

8

5

8

7

83 3 3 3 ( ) ( ) ( ) ( )

=( )

( )

( )

( )

1

4 3

1

4 1

1

4

1

41

k k

k k k

.01065796470989861952... 2 5 2 33

24

5

43

9

82

81

16 ( ) ( ) ( ) ( ) ( ) ( )

= H

kk

k

14

1 2( )

.01077586137283670648... 13

8

2

4

3

12

3 1

3 1 8

3 1

3 1

log loglog

=1

12

1

12

1

12

1

121

1221 1k k

hgk

k

kk

k

( )( )

.01085551389327864275... log2131

192 6 52

dx

x x

6 .01086775003589217196... 1

256

1

4 13

3

40

1

( ) log

x dx

x

... 60

49

3600

1

2 1 2 2 2 6 2 70

( )

( )( )( )( )

k

k k k k k

... 9

43 5

17 5

2

( ) ( )k kk

= 1

1 22 5 15

1 1k k kk

k k( ) ( )( )

.01105544825889466389... 11

1536

1

4

3

4

1

2 33 3

40

( ) ( ) ( )

( )

k

k k

1 .0110826597832979188... 1

2 1 255

11 ( )

( )k k

kk k

k

1 .011119930921598195267... 1

22 2cosh cos

=

sin sin sin

( )sin

( )/ /1 11 2

2

1 2

21 4 3 4 i i

= 114 4

1

kk

... 3 12 2 9 31

12

130

log log( )( )k kk

.01121802464158449834... csc

( )cos cos sin sin

1

4 11 1 3 12

2 2

ee e

1

=1

14 41 kk

... ( )jkkj

3 115

... 1

36 6

1

8

23

0

( )k

kk

k k

k

... log log

( )3

2

2 2

36 r

... 32 2 8 2 4 3 2 4 51

2 6 51

log ( ) ( ) ( ) ( )

k kk

=( )k

kk

5

21

... 720 7 11

6

0

( )( )

x

e edxx x

... 720 7 17 ( ) ( )( )

...

3 2 3 1 1 1120

10

120

dx

x

x dx

x

... 21

71

/k

k k

...

sinh2

1

2 4

2

21

k

kk

=

1

02)1(

)log2cos(dx

x

x GR 3.883.1

... 117

2 7k ks ds

k slog( )

.01175544134736910981... coth

2

12e

...

1

125

1

5 12

2

50

1

( ) log x dx

x

.0118259564890464719... 2

1144

49

864 1 2 3 4

H

k k k k kk

k ( )( )( )( )

... 3 3

2 22 2 4

1

1 2

2 1

31

( )log

( )

( )( )

k

k

k

k k

... 3 2

6 4116

52

27

log x

x xdx

.0121188016321911204... 7

5760

5

192

93

256

1

1

4 2

2 42

( )

( )

k

k k

... e

k

k

k

2

0

7

32

2

5

( )!

.01227184630308512984...

256 162 20

dx

x( )

.012307165328788035744...

1

0

1

0

1

0222

333

132

)3(3

8

1dzdydx

zyx

zyx

.01232267147533101011... 8 1 .01232891528356646996... 5 2 6 3 ( ) ( )

.01236617586807707787... 4 2

2 41

30

768

1

2

1

4 1

( )kk

J373

.01243125524701590587... 18 13

32

9 2

32

13

128

G

log

J385

.0124515659374656125... 80

3

8

45320 2 48 3 560

1

2 1

2 4

4 41

log ( )( )k kk

.01250606673761113456... 1 4 4 8 6 3 12 52

3

61

( ) ( ) ( ) ( )( )

k

kk

.01256429785698989747... 160 44 215

24 80 2 20 3

1

2 13 41

( ) ( ) log ( )( )k kk

.01261355010396303533... 1

12 42 ( )kk

.012665147955292221430...

022 18

1dx

e

exx

x

1 .01284424247707... W3

.01285216513179572508...

( )log

66

1

k

kk

... 1

1536

3

4

1

4 33

40

( )

( )

kk

... 31 3

108

1

1 2 3

2

21

k k k kk ( )( )( )

... 3 3

44 2

24

13

4

1

1 2

2 1

3 21

( )log

( )

( ) ( )

k

k k k k

6 .01290685395076773064... 5

81 1

4 3

60

log x dx

x

... 209

225

1

2 5

1

21

Gk

k

k

( )

( )

1 .01295375333486096191... 1

23 1

21

2

1

16 4 10

log log log( )

ii

ii

kkk

... 110

2

... 1

216

4

3

5

62 2

2

6 31

( ) ( ) log

x dx

x x

1 .01321183515... ( )( )

1 12

1

k

k

k

k

... 8

1 .01330230301225999528...

0 2

12sin

2

1

2

1sin

1cos45

6

kk

k

... 7 3

4

7054

3375

1

8

7

22

2

8 61

( ) log( )

x

x xdx

... 1

43 3 12 2 2

1

12

1 3

30

( ) log( )

( )

k

k

k

k

=49

3600

1

2 1 2 2 2 6 2 70

( )( )( )( )k k k kk

...

2 4 2

4132

3

128

21

163 1

2 3

( )( )

k

kk

... ( ) ( )( )

2 6 465

8

1 4

1

2

42

k k

k kk

= ( ) ( )

1 23

1

k

k

k k 1

... cos sin( )!( )

1 11 1

4 2 10

e k kk

... 1

512

9

8

5

82 2

2

6 21

( ) ( ) log

x

x xdx

... iLi e Li e

k

ki i

k2 2 21

sin2

e

... e ie i /2

...

2

4 2coth ( ) ( ) 1

= ( ) ( )

1 2 4 111

16 4

2

k

k k

kk k

... dxxx

x

0

222

)3)(1(

log3log

12

3log

... 3 3

256

1

4 30

( ) ( )

( )

k

k k k

.01417763649649285569... 2 3

27

7

18 12 31

dx

x x( )

.0142857142857142857 =1

701

162

5

kk

2 .0143227335483157... F

kk

k !

1

5021 .01432933734541210554... 127

240 1

8 7

0

1

log x dx

x GR 4.266.1

3 .0143592174654142724... 1

1 Fkk

.01442468283791513142... 3 3

250

2

61

( ) log

x dx

x x

... e k

k

( )

1

1

1

1 .01460720903672859326... e EikH

kkk

k

12

1

2

1

2

2

21

21

log

!

.01462276787138248... ( )2

142

k

kk

1 .01467803160419205455...

4

4196

7 4

84

1

2 1

( )( )

( )kk

AS 23.2.20, J342

... 27

28

8

)3(7

=

1

0

1

0

1

0222

444

1dzdydx

zyx

zyx

... 1

1728

11

12

5

12 12 2

2

61

( ) ( ) log

x dx

x

... 2 2

12

2

1

k

k

k

k

k

ke

k

( )

!/

...

1

12

332log

4

331

8

27log

4

1

kkkH

... e

erf e x x dxx4

12

0

( ) sinh cosh

... 2 3

44

Berndt 8.17.16

... 17

16 48 180

3

4

1

2

2 4

41

( )

( )k kk

.01521987685245300483... ( ) ( ) ( ) ( )( )

4 3 6 3 5 71

3

71

k

kk

... 2 2

40648

2 1

6 3

( )

( )

k

kk

... log ( )

( )( )

2

8 16

1

8

1

4 2 4 4

1

1

k

k k k

...

2

3 21

5

43

1

1 2

( )( ) ( )k k kk

... j6 J311

...

( )36

3

1 3

6

3 3

2

1 3

6

3 3

2

i i i i

= ( ) ( )

1 3 3 111

16 3

2

k

k k

kk k

... 45 5

484 3 504 2 462

1

1

1

6 61

( )( ) log

( )

( )

k

k k k

2 .01564147795560999654... ( )8

... 1

61 ( !)kk

... 2 6 42 4 252 2 4621

16 61

( ) ( ) ( )( )

k kk

... 13

1 k kk

...

1

8192

1

4

1

16 42

30

( )

( )kk

1 .0158084056589008209... log

log log( )2

2

1

2

2 1

2

2 1

2

2

21

k

kk

... dx

x52 1

.01593429224574911842...

( )4 1

4 14 11

kk

k

1 .01594752966348281716... 104

98415

6

6

G J309

.01594968819427129848...

1

3456

1

3

243 32 3

124416

1

3456

5

32

32

( ) ( )

= 1

12 8

13 3

1728 2592 331

3

( )

( )

kk

.01600000000000000000 = 2

125

2

61

log x

xdx

1 .01609803521875068827... 1

41 k k

k

.01613463028439279292...

254

15log2log

xx

dx

.01614992628790568331... 92

32 3

1

1

2

2 32

( )( )

k

k kk

= ( ) ( )

1 31 2

1

k

k

k k 1

.0161516179237856869... 10163

6 4

2

1

ek

kk

( )!

.0162406578502357031... 31 5

8

7 3

32

2

48 2 1

2 4

41

( ) ( ) log

( )

H

kk

k

2 .01627911832363534251... 2

71

k

k k k!

1 .01635441905116282737... 1

51 k kk !

.0163897496007479593... 1 22 16

21

2 1

3

32

Gk k

k

k

log

( )

( )

... 2

6

2 2

2 2

1

2

sin sinh

= 11 4 26 2

2

1

1k kk

k

k

k

( ) ( ) 1

6

.0164149791532220206... 6 2 3 4 5 ( ) ( ) ( ) ( ) ( )

= 1 1

16 17 6

26

1 1k k k kk

k k k

( )

( )

.01643437172288507812... 7 3

512

1

8 4 31

( )

( )

kk

.0166317317921115726... ( ) ( )

13

2

k

k

k

k

.01684918394299926361... log

arctan7

6

1

2

3

6

3

3

5

3 5 22

dx

x x

.01686369315675441582... 11 5 16

2

1

1k kk

k

k

k

( ) ( ) 1

.01687709709059856862... 6155 63

1080

4 3

5 41

log x dx

x x

.01691147786855766268... 2

2112

29

36

1

1 2

3

k k k kk ( )( ) ( )

.0170474077354195802... 1945 6 1

66

( )

( )

... 13 3

216 324 3

1

432

2

3

1

6 2

32

31

( )

( )( )

kk

...

6

16

1 3

2

1 3

2coth cot

i i

12

2 2 32

2 3i icot

= 1

11 66

2

1

1kk

k

k

k

( ) ( ) 1

... 3 2 2

8

3

2 12 3

0

dx

ex /

... 6 22

4 221

2 32

log( )

k k

k

kk

k

...

6

9456 ( )

... 1

120 1

1

5

5

2

5

2

log

( ) !( )( )k

k km

k m

1 .01736033215192280239... 1

22 22 k kk

1 .01737041750471453179... sinhcosh cos

4

3 11

3 62

kk

... 18 2 16 2

2 1k kk

k k

( )

... sin1

...

180 , the number of radians in one degree

... tan1

... 17 1 16 1

2 1k kk

k k

( )

... ( )

log( )6 1

11

6

2

k

kk

k k

... ( ) ( )( )

3 3 433

16 3 41

k

kk

... 11

12 2 3

3

2

1

16 16

2 1

tanh ( )

kk

k k

... ( ) log( )5 1 1 1

1

5

2

k

k

k

kk k

... 6

2

3

2

1

2

2

6

62

sin

i

k

kk

.01765194199771948957... 2

21

8

144

1

12 3

G

kk ( )

.01772979515817195598... 1

6

11 2

18

3

4 1 21

log log

( )( )(

dx

e e e ex x x x 3)

... 1

144

1

12

5

12

7

12

11

121 1 1 1 ( ) ( ) ( ) ( )

=( )

( )

( )

( )

1

6 5

1

6 1

1

2

1

21

k k

k k k

... 15 1 16

2 1k kk

k k

( )

... sinh( )

32

3

2 1

3 3 2 1

0

!

e e

k

k

k

AS 4.5.62

... ( )

( )arctan log

1

2 2 1 93

1

3

10

91

1

1

k

kk k k

... ( )5 ... e4

.018355928317494465988...

16)1(

)1,6()4()3()5()2()7(3k

k

k

HMHS

... 7

82

4

14 2 16 2

2 1

( ) coth ( )k k

kk k

.01840776945462769476... 3

512 42 30

dx

x( )

... 7 3

16 192 2 1

4

41

( )

( )

k

kk

...

2

1)()(k

kk

... 5

33

1

2

1

4 2 1 2 31

arctanhk

k k k( )( )

... ( )

( )

3

64

1

4 31

kk

... loglog

23

2

1

8 5 32

dx

x x

.01891895796178806334... log log2

3

3

4

1

30 4 3

1

36 622

k kk

=( )k

kk

1

62

1 .01898180765636222260... ( )4 1

2

1

215

1

k

k kkk k

.01923291045055350857... 2 3

125

2

61

( ) log

x

x xdx

1 .01925082490926758147...

( )

( )

( )5

6 61

k

kk

Titchmarsh 1.2.13

3 .01925198919165474986... e d

e xx

2 2 4 1

20

( )

x

.01931537584252291596... 3

4

2

3

2

22 3

3

2

1

3

1

8

2 2

2loglog

log loglog

Li

= ( )

( )

1

2 4 22

k

kk k k

.01938167868472179014... 112

3 3

2

1

2

2 1

31

( ) ( )

( )

k

k

k

k

.01939671967440508500... G k

k

k

k2 16 128

1

2 1

3 1

31

2

( )

( )

.01942719099991587856... 2

5

7

8

1 2 1

2 42

( ) ( )!

( )!

k

kk

k

k

!

.01956335398266840592... J3 1( )

.01959332075386166857... 14

96

2

32

1

2

1

2 12

1 3

21

log( )

( )Li

k

k

k

kk

.01963249194967275299... 4

33

1

61 3

3 3

21 3

3 3

2

( ) i

ii

i

= 13 3 1

1

13 16 3

2 13

2k kk

kk k k

( ) ( )

=1

13

1

a non trivialegerpowerint

306 .01968478528145326274... 5

.01972588517509853131... 1

9 108

1

3 3

2 1

21

( )

( )

k

k k

.01982323371113819152...

4

4190

17

164 3

1

2

( , )( )kk

.019860385419958982063... 3 2

4

1

2

1

64 43 21

log

k kk

[Ramanujan] Berndt Ch. 2

.0198626889385327809... 3

8

7

1920

4

5 31

dx

x x

.01990480570764553805... 3

64

3

32

3

41

31

2

1

log ( )k kk

k

k H

1 .0199234266990522067... log

log ( )5

2

1

5

1 5

25

r

1 .0199340668482264365...

2 2

2 22

2

16

5

8

1

12 2

k

kk k k

k k( )( ) ( )2

.02000000000000000000 = 1

50

... 1

22 1 30 k

kk

k

( )

.02005429455890484031... 2

9

7 2

241

1

1 141

42

log

log( )

logx

dx

x

x

x

dx

x

.02034431798198963504... 10

9

1

3

2 1

2 1

1

2 31

2

( )!!

( )!!

k

k kk

J385

.02040816326530612245... 1

49

1 .02046858442680849893... 62 5

63 61

( ) ( )

a k

kk

Titchmarsh 1.2.13

.02047498888852122466... 4 5 4 4 3 12

2

51

( ) ( ) ( )( )

k

kk

.020491954149337372830...

( )

( )

( )

( ( ) )3

3

6

1 13

14

k

k nk n not squarefree Berndt 6.30

1 .0206002693428741088... 1

9

8

9 190

csc

dx

x

.02074593652401438021... 7 3

8 322 2

1

64

1

4

32

2

6 21

( ) log( )

x

x xdx

2 .02074593652401438021... 7 3

8 32

1

64

1

4 1

32

2

40

1

( ) log( )

x

xdx

.020747041268399142... B

kk

kk

2

1 2 4( )!

1 .02078004443336310282... 13 3

27

2

81 3

1

54

1

3

1

3 2

32

31

( )

( )( )

kk

Berndt 7.12.3

33 .02078982774748560951... 416

3

3

2

2

0

( !)

( )!

k

k

k

k

.020833333333333333333 = 1

48

1 .020837513725616828...

2 3 11

21

1

2

1

2 5 31

( )k kk

= ( )2 3

21

kk

k

1 .02083953399112117969... 1

2

1

2

1

2

1

4 24 40

cos cosh( )!

k k

k

.020839548935264637084... 1

6 4 2 2 2

4 1

41

cot coth

( )kk

k

... ( )

!

k

kk

1

... 3

22

50125 5

1 25 1

csc cotlog

x

xdx

=

4 2 25 3 5

125 5 5

3

5 2

/

... 197

108

3 3

2

2

5 41

( ) log x dx

x x

... 1

8

1

23

1

2 2

2

20

1

, ,log

x

xdx

=1

2

1

2

1

23 3Li Li

...

9

1

18 6 3

9 3

6

1

18 6 3

9 3

6

i i i i

=1

3 1 31 ( )k 1k

... ( )k

kLi

k kk k

1 15

25

2

1

=691

3276011 ( )

... 1

47

... 1

3

2

3

3

3 1

1

31

21 1

( )

( )

( )

k

k k

kk

k

... ( )

logk

k kkk k

1

4

1

41

1

42 1 k

... 7

1 .02165124753198136641... 41

16 2 10

arctanh1

4

kk k( )

... 1

46

... 42 26 22

3

1

logk Hk

kk

...

k

k

kkk

2

)12(32

1

22

1arcsin22

0

... 7

30720

4 3

51

log x dx

x x

=1

45

... 2

218

3

2

29

16

1

1 3

( )

( )( )k k kk

...

2

54

2

9

1

9 3 3

5 2

2

1

9 3 3

5 2

2

i i i i

1

18 6 3

5 2

2

1

18 6 3

5 2

21 1i i i i

( ) ( )

= 1

11 3 1

3 22 1k

k kk k

( ) ( )

...

12 )32)(22(4

)2(

2

)3(

12

1

kk kk

k

... 4 211

4

1

1 2

1

1

log( )

( )( )(

3)

k

k

k

k k k

... log log

( )

2

3

5

24 1 40

1

x

xdx

=1

44

... 11

42 4

1 1

1 26 42

41

( ) ( )( ) (k k k k k )k k

... 17

16

3

22 4 4

0

4

log sin tan/

x x d

x

... 9

8 12

3 2

4

2

2

1

2

2 2

4 22

log log

( )kk k k

.02291843300216453709... 3

23

13

8

1

9

2 1

932 1

log( )

1

k k

k

kk

k

... 1

1542 kk

1 .0229259639348026338... 1

2 1 244

11 ( )

( )k k

kk k

k

... 50 43

128

28 2

128

71

1536

2 1

2

1

2 61 3

2G k

k kk k

log ( )!!

( )!!,

...

64

2 2 22

1

2 1 22

2 21

sin sec(( ) )kk

... )(,1

2log12

log

2sofzeroa

Edwards 3.8.4

... 31 5

8

7 3

32

7 3

4

2

48 48 42 2

2 4 4 ( ) ( ) ( ) loglog

=H

kk

k ( )2 1 41

... log k

kk2

2 1

.023148148148148148148 = 5

216

1

5

1 3

1

( )k

kk

k

1 .02316291876308017646... 2

2 21

2 3

6

1

12 1

1

12 11

( )

( ) ( )

k kk

=log x dx

x120 1

.02317871150152727739... 233

1920

2

32

1

1643

coth

kk

... 1

43

... 3 2

3148

2

4

1

4 2 1

2

log

( )kk k

k

k

= arcsin logx x

xdx

0

1

.02334259570291055312... 3 2 3 3 2 3 2 31

2

0

( ) ( ) ( )logx x

e edx

x x

.02336435879556467065... 5

36

2

6

log

=( )

( )( )( )( )

1

2 1 2 2 2 4 2 50

k

k k k k k K ex 107b

.02343750000000000000 = 3

128

1

3

3

1

3

51

( ) logk

kk

k x

xdx

... 2

2 4

2

21

sinh

k

kk

.02351246536656649216...

3

5

27

1

4

1

381

4

27

)3(26 )2(3

=

2

3173

2

3134

)3(6

1

381

8

3

)3(233

3 iLii

iLii

i

= dxxx

x

136

2log

... 2 328567

1200072

2

8 71

( )log( )

x

x xdx

=1

42 3 B

... 2

20

6 5112

115

144

1

5

( )

( )

k

k k

dx

x x

...

2

2 211

1

21

1

2 1

cosh( )kk

2

... ( )jkkj

2 114

... 1

162 6k k

s dxk slog

( ( ) )

... ( ) ( )

( ) /

1 2 1

21

1

21 20 1

3k

k kk

kk

k HW Thm. 357

... 16 2 4 2 2 3 41

2 5 41

log ( ) ( ) ( )

k kk

=( )k

kk

4

21

... 21

61

/k

k k

... G k

kk8 64

1

256

1

4

1

6144

1

4

2 1

4 1

22 3

2

40

( ) ( ) ( )

( )

=G

8 64 128

7 3

32

1

6144

1

4

2 33

( ) ( )

=1

41

... 1728

2035)3()5,3(

=

1

0

1

0

1

0

444

1dzdydx

xyz

zyx

3 .02440121749914375973... ( )

!

/k

k

e

kk

k

k

3

0

1

31

.02461433065757607149... 4 3 3

16

2

5 31

( ) log x dx

x x

1 .02464056431481555547... 2 1 272

64 2 12 2 22 32 1

24

23 2

1

( log ) log log ( )( )

H

k kk

k

.02475518879810900399... 61 6

72

1

1 2

2

21

3

k k k kk ( ) ( )( )

7 .02481473104072639316... 5

.0249410205141828796... log( )

5

6 6 62

k

kkk

.02500000000000000000 = 1

40

... ( ) ( ) ( )2 3 43

2

15 4

2

k kk

= ( ) ( )

1 41

1

k

k

k 1

... 1

4 2

=1000

3937 = meters/inch

.02553208190869141210... ( )3 1

8

1

8 113

2

k

kkk k

... 4430

21871

21

8

( )k

kk

k

=1

39

... 23

7

3

2

1 32

1

1

arcsinh

( )k k

kk

k

k

... 2

22 2

18 2 4 2

1

2

1

2 1csc cot

( )

kk

J840

... ( ) ( )

( )6 7 1

71

k

kk

HW Thm. 290

.02584363938422348153... k

k kx d

k

1

162 5

6

log( ) x

.0258653637084026479... 2 5 2 3 4 3 2 42 4

1

( ) ( ) ( ) ( ) ( ) ( )( )

H

kk

k

.02597027551574003216... 1

2 162 1

1

8 1 8 11

( )

( )( )k kk

GR 0.239.7

1 .02617215297703088887... 8

7

8 180

csc

dx

x

.026227956526019053046... 161

900 6 0

1

G

x x dx arccot5 log x

=1

38

.02632650867185557837... 1

2

7

21

2

12

2

0

e

k

kk

k

k

( )( )!

... 1233

2

1

ek

k

k

k

!

.02648051389327864275... log( )

22

3 1 41

5 42

dx

x x

dx

x x

... 3 1

3164 2

1

2 1

G k

k

k

k

( )

( )

... 3 3

4

7

8

1

3 30

( ) ( )

( )

k

k k

=

1

0

1

0

1

0

222

1dzdydx

xyz

zyx

...

1 2

)2(

2csc

2log

kk k

k

=

122

11log

k k

... 4 7 2 5 3 46

1

( ) ( ) ( ) ( ) ( )

H

kk

k

... ( , )

.02681802747491541739...

3

2

6

7

216

1

18

)3(13

381

52 )2()2(

3

=

2

3132

2

314

)3(3

1

2

)3(

381

102 33

3 iLii

iLii

i

=

125

2log

xx

dxx

... 1

81

1

3 13

3

30

1

( ) log

x dx

x

=1

37

... Gk

x dx

x x

k

k

8

9

1

2 5 20

6 41

( )

( )

log

... ( )

( )!cosh

k

k k kk k

1

21

1

2

1

2 2

... 2 4

4164 768

1

2 2 1

k k

kk

( )

( ) J349

... 3

2

5

324 2

0

4

sin tan/

x x dx

...

1 .0275046341122482764... B

kk

k

2

0 2 1( ) !

.02755619219153047054...

( )

( )

log

( )

log5

5 1

1

55 51

p

p

k

kp prime k

3 .0275963897427775429... cosh( )

!

e e

k

k

k2 20

GR 1.411.4

.02768166089965397924... 8

289 40

x x

edxx

sin

... ( )k

kLi

k kk k6

26

1

1 1

... 3

64 2

1

4 3

1

4 1

4 1

3

1

31

( )

( )

( )

( )

k k

k k k J326, J340

.02777777777777777777 = 1

36

1

2 1 2 2 2 4 2 50

( )( )( )(k k k k )k

K ex. 107c

= k

k k k kk ( )( )( )( )

1 2 30 4

... ( )

( )

( ) ( )

( )

1

4 4 1 4 42 1 1

k

kk k kk

k

k

k

k k

1

.02788022955309069406... 115 5

16

1

2 50

( ) ( )

( )

k

k k

... 7 183

2

0

ek

k kk

!( )

... 7 172

4

0

ek

kk

( )!

... 7e

... 1)2(sinh

cscsinlog2

11

1

kk

ke

e k

.02817320866780299106... 3 3

128 1

2

51

2

40

( ) log

x dx

x x

x dx

e x

... 1

...

8csc

8

3sinlog222log8

8

7cot

16

1

=

21log222log8

8

7cot

16

1

=

12

2 864

1

8

)(

kkk kk

k

.02839399521893587901...

145

21

2

cos

1cos

)1(cos1

51cos4

1cos21

kk

k GR 1.447.1

... 3

2

4 1

3

1

1

e

k

k

k

k

( )

( )!

=1

35

...

( )log

55

1

k

kk

... ( )ck

kLi

kkcc

kc

1 12

122

21

...

1

31

2)!1(

)1(

4

132

kk

k

k

k

e

... 3 3

22

4 1

2 3

41

( )log

log

( )

x

xdx

... log log11

30

12

x

x dx

= 122

34 2

1

6

7

6

2

3

1

72

2

3

7

61 1 2 2

log ( ) ( ) ( ) ( )

.0290373315797836370... 1

1000

9

10

2

5 12 2

2

51

( ) ( ) log

x dx

x

.0290888208665721596...

108 92 20

dx

x( )

3 .02916182001228824928... sinh

( )

3

7 31

3

2 21

kk

.02929178853562162658... 1

48

1

16

1 2

420

e k

k k

k

( )

( )!

1

... 4 1 2

2

k

k

k( ( ) )

... 1

34

.029524682084021688... c34 2 21

2412 2 8 3 3 2 6 4 ( ) ( ) ( ) ( )

Patterson ex. A4.2

.0295255064552159253... 7 3

4

56

272

1

2 1 32

( )

( )

kk

= log2

6 41

x

x xdx

= x x

xdx

4 2

20

1

1

log

GR 4.261.13

.02956409407168559657... 1

4

1

36

5

6

1

6

1

3 21 1

1

21

( ) ( ) ( )

( )

k

k k

= 1

36

5

6

4

31 1

6 31

( ) ( ) log

x dx

x x

...

0 2)12(22

1

8

1,

2

3,

2

3,1,1,

2

1

k k kk

kHypPFQ

...

12 )32)(12(4

)2(

8

)3(9

6

1

kk kk

k

1 .02967959373171800359...

3 2

40

1

1

4

163

41

/

log

x dx

x

.029700538379251529... 2

3

9

8

2 1

2 42

( )

( )!

k

k kk

!!

...

3 33

3 3 1330

log

x dx

e x GR 3.456.1

=

log x dx

x1 330

1

GR 4.244.2

.0299011002723396547...

2

214

39

16

1

1 2

k k kk ( )( )

J241, K ex. 111

... 3 2

2

2

2

5

4 2 1 2

2

1

log log

( )( )(

3)

H

k k kk

kk

.0300344524524566209... 2

27

3

21

2

1

log ( )

k kk

k

H k

2

... log ( )( )

6

5 51

52

kk

k

k

k

1 .0300784692787049755... ee erf k

k k kk

3

20 1

1

21

2

2 1 2 11

, ,( )!!

( )!! !(

)

2 .0300784692787049755... e erf k

k

k

k

1

2

4

2 10

!

( )!

.03019091763558444752... ( )31

2

1

21 1 i i

= )2()2(2

1)3( ii

=

1

1

235

1)32()1(1

k

k

k

kkk

... 3 2

4 401024

x dx

e ex x

.03030303030303030303 = 1

33

1 .03033448842250938749... 6 3

7

( )

... e

e

...

( )( )

3

42

2 2 c Berndt 9.27

1 .03055941076... j5 J314

.03060200404263650785... 1

64

1

4

1

16

1

4

1

2 16

1 2

1

cos sin( )

( )!

k

kk

k

k

2 .03068839422549073862...

2 2

322

1 2 35 5

1

2

( )( )

k k

k kk

= ( ) ( )( ( ) )

1 12

2

k 1k

k k

.030690333275592300473...

12

11

1

2

log ( )( )

kk

k

k

k

... dxee

xxx

022

34

18

3

240

.03095238095238095238 = 13

420

1

2 1 2 5 2 90

( )( )(k k k )k

.03105385374063061952... 132

1

2 1

3 1

31

( )

( )

k

k k

.03111190295912383842... 33

48 48

9 3

24

1

2

2

31

( ) ( )

( )

k

k k k

... 15 5

8

45 3

2140 2 126

1

1

1

5 51

( ) ( )log

( )

( )

k

k k k

... log

( )( )

( )(

2

4

1

61

1

1 22

k

k k kk )

5 .03122263757840562919... 2 2

2 1

1 22 1

1 1

k

k k

k

k k

k

( )

( )!sinh

... 2 1

216

2 2 31

1 2

log( )

( )( )

k

k

k

k k

... 1

32 84

1

16

1

21

csch

( )k

k k

.03125002980232238769... 1

2

5

32 151 1

k

kk

k

k

( )

.03125655699572841608... 23

2 2 81

2

22

3

log Li

1 .03137872644997944832... 1

51 ( !)kk

.03138298351276753177... 12635

3 9

1

1

2 4

5 51

k kk ( )

1 .03141309987957317616... cosh( )!

1

4

1

2 160

k kk

... 1

642

5

82

9

8

1

64

5

8

9

81 1 , , ( ) ( )

= ( )

( )

1

4 1

1

21

k

k k

= log x dx

x x6 21

... 12

2

1

11

12 2

2

logsinh

coth log

k kk

= ( ) ( )

11

2 11

k

k

k

kk

.032067910200377349495... 1

6 64

5 2

12 1 1

2

41

log log

( ) ( )

x

x xdx

... 527

7350

2 2

35

1

2 7

1

1

log ( )

( )( )

k

k k k k

=1

32

2

51

log x

xdx

... 123

2

29

6

1

2 2 6

1

0

log( )

( )

k

kk

k

k

... 1

3

=1

31

... )6(48

203)3(3)1,2,3( 2 MHS

... ( ) ( ) ( ) ( )5 3 3 2 4 5 2 15

=1

12 51 k kk ( )

... 5

2 4

1

1 2

2

2 21

k k kk ( ) ( )

... log

( )

2

4 2 2

2 1

8 2 102

G k

k kkk Berndt 9.6.13

1 .0326605627353725192...

15

155 )13(

1

)13(

11

243

)3(242

kk kkG

J309

1 .03268544015795488313... 1

31 k k

k

.03269207045110531343... 3

4

5

6

1 2 1

2 32

( ) ( )!

( )!

k

kk

k

k

!

... 96 646

0

dx

x

.03283982870224045242... 19

72

2

3

1

2 4

1

1

log ( )

( )( )

kk

k

H

k k

... 2

61300

log x dx

x x

.03302166401177456807... J

k kk

k

k1

0

4

41

4

1

( )!( )!

... 365

48

1

4 1

1

0

log( )

( )(

3)

k

kk

k

k k

... 7 3

16 192 2 1

4

41

( )

( )

k

kk

... 2

61

k

k k k!

...

3

3

2 6

1

61 3

3 3

21 3

3 3

2

2

( ) ( )i

ii

i

= ( ) ( )

1 3 2 111

15 2

2

k

k k

kk k

.0333333 = 1

30 2 4 B B

1 .03348486773424213825... 1

41 k kk !

... 3

30

.03365098983432689468... 1

27

1

3

13 3

81

2 3

7291

3

( ) ( )

= 1

1626

1

3

1

3 31 2

31

( ) ( )

( )

k

kk 1

.033749911073187769232...

1

02

6

)1(

)1log(2log3

34

5dx

x

x

.03375804113767116028... 5 2 3 4 51

1 51

( ) ( ) ( ) ( )( )k kk

= 1

6 52 k kk

.03388563588508097179... ( ) ( )k k

kk

0

1 2 1

1 .033885641474380667... pq k

kk

( )

21

.03397268964697463845... 3

2 12

2

2

2

2

1

2

2 2

3

log logLi

.03398554396910802706... ( )

( )!sinh

2 1 1

2 1

1 1

1 2

k

k kk k

k

.03399571980756843415... J Fk k

k

k4 0 1

0

21

245 1

1

4( ) , ,

( )

!( )!

LY 6.117

.03401821139589475518... 1

144

5

12

11

12 11 1

61

( ) ( ) log

x dx

x

.03407359027997265471... log

sin tan/2

2

5

164

0

4

x x d

x

1 .03421011232779908843... HypPFQk

k kkk

1

2 1 12 2

1

4

2 1

16 102, ,

, , ,( )

... 3 3

22 2 3

1

36 31

loglog

k kk

J376

... 1

2

11 1

12

1

ee

erf erfix

dx

sinh

... 4

2 34

2 2 11

3

4

2

20

1log( !)

( )!( )

k

k k

k

kk Berndt Ch. 9

... 7

6

7 170

csc

dx

x

... 1

29

... 1 1

3

1

3

1

1e k

k

k

( )

( )!

... 12 2 21

1 340

log( )(

)

k kk

.034629994934526885352... 3

2 122 2

3 3

4

12

4 32

log

( ) ( )k

k k k

... 1

1024

7

8

3

8

1

4 12 2

1

31

( ) ( ) ( )

( )

k

k k

... sin

arctan

x

xdx

0

1

2 .03474083500942906359...

22

3coscosh)(sinh

= sinhcosh cos cosh cos

4

1 1 3 1 1 33

= 11

61

kk

Berndt 4.15.1

1 .03476368168463671401... 1 11 1

1

2 12

2

2

e

k k e

k

kk

k

kk

k

k

/

/ ( )!

( )

=

Re( ) 2 1

1

k

ikk

2 .03476368168463671401... 1 11

11

1

2 11 2

e

k k e

k

kk

k

kk

k

k

/

/ ( )( )!

( )

.03492413042327437913... Ai ( )2

... 21

2

1

2 1 24 40

4

1

Lik

Hk

k

k

kk

( )

( )

...

1

2

1

4

1

2

1

2

1

2

1

2

i i i i

= ( ) ( )

1 4 1 111

15

2

k

k k

kk k

.035083836969886080624... 1

21 1

2

log( )

k

kkk

... ( ) ( )( ) ( )

( ) ( )( )

( )2 34

4

3 6

42 3

3

22 5

2

=H

k kk

k ( )

1 51

... 2 3 15

12 csc

AS 4.3.46, CGF D1

.03532486246595754316... 11

32

1

1

2

2 32

( )

( )

k

k k

... 1

9

...

3

4

1

2 2 1 2 2

1

1

( )

( )( )(

k

)k k k k J244, K ex. 109c

...

1

125

4

52 ( )

.03561265957073754087... ( )

( )

5 1

5

...

1

12

)12()1(

2

1

kk

k kii

=1

28

... 1

55

1p

k

kk

p prime k

( )log ( )

... 2

25 5

1 2 1 22

21

3

21

( ) ( )!

( !)

( ) ( )!

( !)

k

k

k

k

k k

k

k k

k

1 .03586852496020436212...

1

4)14()4()14()24(k

kkkkk

... 1

11 55

2

1

1kk

k

k

k

( ) ( ) 1

... 2 4 2

4132 384

7 3

16 2 1

( )

( )

k

kk

.03619529870877200907... 2 7 2

81

1

12 31

log

log( )x

dx

x

.03635239099919388379... 1

2

1

2

1

2 2 12 11

arctan( )

( )

k

kk k

... k

kk

k

k

k6

2

1

111 6 1

( ) ( ) 1

... 1

5

6

5

1

5 611

log( )

H nk

kkn

... 2 2 23

2

1

2

log

... log( )

log4

5 5 51

1

5

1

52 1

k

k kkk k k

1 .03662536367637941437... 1

36

1

6

1

6 5 11

21

60

1

( )

( )

log

k

x dx

xk

... 2 14

2 1

0e k

k k2

k

( )

!

...

1

1

27

2

1)27()1(1 k

k

k

kk

k

... 3 1

2 31128

3

32

1

2

1

4 1

( )

( )

k

k k

...

032 )4(256

3

x

dx

...

12 )22)(12(4

)2(

4

)3(7

4

1

kk kk

k

... dxxx

li log2sin1

2log225

1 1

0

... ( )5 ... 6

= 1

27

...

1235

1)38(1

kk

kkk

.037087842300593465162... 1

3

3

27 131 1

k

kk

k

k

( )

...

1225

1)27(1

kk

kkk

... ( )

log( )6 1 1

11

6

2

k

kk k

k k

2 .03718327157626029784...

21

3

5

32

...

03

)2(

)316(

1

16

3

8192

1

k k

... 9

4

3

2

7

8

1

2 122

log( )

( )

k

kk k k

... coth22 2

2 2

e e

e e

=1

2

16

2 20

k

kk k

B( )!

AS 4.5.67

.03736229369893631474... 1

2

3

64

1

4 1

2

2 31

( )kk

J373, K ex. 109d

... 1 2 4 32 4

1

( ) ( )( )

k

kk

...

13

)2(3

)912(

1

4

1

3456

1

432

)3(7

1728 k k

...

1215

1)16(1

kk

kkk

...

21

1!

1)5( 5

k

k

k

ek

k

... ( )

log( )5 1

11

5

2

k

kk

k k

... 3 2

4

19

20 8

1

4 4 11

1

42

1

2

log

( )( )

( )

k k

k

k

kk

k

.03756427822373732142... dxe

xdx

xx

xx

04

2

15

2

1

log

32

)3(

.03778748675487953855...

032 )3(348 x

dx

... 6

...

13

)2(

)69(

1

3

1

1458

1

k k

... 16 29

3

1

4 20e k

k

kk

( )

( )!

... 3 3

42 2

9

4

15 3

2

( )log

( )

k

k k k

=

13

1

)2()1(

)1(

k

k

kkk

.037901879626703127055... 1

2

2 2

3

1

27 331

log ( )k

k k k [Ramanujan] Berndt Ch. 2

...

12

51)5(

1

1

kk

kk

... ( ) log( )4 1 1 1

1

4

2

k

k

k

kk k

...

1

2)12(

812cos

7

1

k k

...

13

)2(

)58(

1

8

3

1024

1

k k

...

0

sinlog4

2log

22

1dxxxex x

1 .0381450173309993166... k

kk

5

52 1

.0381763098076249602... 80 4 48 2 6 31

2 12

3 31

log ( )( )k kk

.03835660243000683632... dxx

x

1

02)22(

)1log(

8

2log1 GR 4.201.14

1 .03837874511212489202...

0

1

1112

)1(

k

k

k

.0384615384615384615 =

0

5cos26

1dxex x

.0385018768656984607... cossin

cos sin cos22

22 1 1

=

1

21

2/32/1 )!2(

4)1()2()2(

k

kk

k

kJJ

31 .03852821473301966466... 3 3

.0386078324507664303...

022 )1)(1(2

1

2

1xex

dxx Andrews p. 87

= 1log4

log1

022 x

dx

x

x

GR 4.282.1

.03864407768699789287... dxxx

x

14

2

)1(

log

36

331

... dxxx 22/

0

23

sin488

... 7

8 162

21 3

2

64 20 8 1

2 2 1

2 4 3 2

42

log( )

( )

k k k

k kk

=k k

kk

3

2 2

( )

...

122

21

02

22

)1(

log

)1(

log)3(3)2(2

xx

dxx

x

dxxx

... 3

)3(

... 1log16

112log2

28

12

1

022

x

dx

x GR 4.282.9

.03886699365871711031... 21

2

1

2

5

41

1

2 2

1

2 11

arctan log( )

( )

1

k

kk k k

... 7 3

216

1

6 3 21

( )

( )

kk

=3929

100800

1

5 81

k k kk ( )( )

... 3

11

!

1)23(

22

1

k

kk

ekk

k

... 1

83 4 2 2 ( ) ( )i i

= 5

8 2

1

42 2

15

2

( ) ( )i i

k kk

= ( )4 1 11

kk

... ( ) log( )3 2 1 1

1

3

22

k

k

k

kk k

... 116

21

163

2 3

2

31

( )( )

k

kk

... 25 5

225 2

11

4

1

4 1

1

1

loglog

( )

( )(

2)

k

kk k k k

... 1

12 32 ( )kk

.03945177035019706261... 1

0

322

log)1log(54

12dxxxx

... 1)2(1

1

4

9log2

2

kk

k

k

1 .03952133779748813524... 2 1cos e ei i

3 .03963550927013314332... 30 5

22

( )