notes on the riemann hypothesis and 1ex abundant...
TRANSCRIPT
Notes on the Riemann hypothesisand
abundant numbers
Keith Briggs
more.btexact.com/people/briggsk2/
2005 April 21 14:27
RH abundant.tex (‘notes’ option) typeset in pdfLATEX on a linux system
RH and abundant numbers 1 of 22
Introduction
F In [1], Lagarias showed the equivalence of the Riemann hy-pothesis (RH) to a condition on harmonic sums, namely
RH⇔ σ(n) 6 Hn+exp(Hn) log(Hn) ∀n
F Robin [3] had already shown that
RH⇔ σ(n)/n < eγ log log (n) for n > 5040
which is probably more convenient for numerical tests
F so to disprove RH, we need to find n with large σ(n)/n
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Notation
F n is always a positive integer; p is always a prime
F pi = ith prime (p1=2, p2=3, . . . )
F harmonic sum for n > 1 is Hn =∑n
i=1 i−1
F sum of divisors is σ(n) =∏
ip
ai+1i −1pi−1 for n =
∏i pai
i , (ai > 0)
F σ(n)/n =∏
ipi−p
−aii
pi−1
F I define ρ(n) ≡ σ(n)/n
F eγ = 1.78107241799019798523650410310717954 . . . , where γ isEuler's constant
F we will deal with n too large to represent in the computer(≈ 10(1010)), but luckily we have its prime factorization whichsuffices
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Superabundant numbers [2]
F n is superabundant (SA) if σ(k)/k < σ(n)/n for all k < n
F if n = 2a2 3a3 . . . mam is SA, where m is the maximal primefactor, then a2 > a3 > · · · > am
F if 1 < j < i 6 m, then |ai−baj logi jc| 6 1
F am = 1 unless n = 4 or 36
F iai < 2a2+1, i > 2
F m ∼ log(n)
F SA numbers, with CA numbers in red:2, 22, 2.3, 22.3, 23.3, 22.32, 24.3, 22.3.5 . . .
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The work of Robin [3]
F Theorem: RH⇔ ρ(n) < eγ log log (n) for n > 5040
F Theorem: independently of RH, except for n = 1, 2, 12:
ρ(n) < eγ log log (n)+[7/3−eγ log log (12)] log log (12)
log log (n)
The numerator in the last term is about 0.6482
F But note lim supn→∞σ(n)
n log log (n) = eγ
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The structure of the set of CA numbers
F Defn: n is colossally abundant (CA) if there exists ε > 0 suchthat
σ(n)/n1+ε > σ(k)/k1+ε ∀k > 1
F Robin [3] has given the following precise description, dueoriginally to Erd ′′os & Nicolas [6]
F we first form the set E of critical ε values:
Ep ≡⋃
α=1,2,3,...
{ logp
(1+
1∑αi=1 pi
)}
E ≡⋃p
Ep
F we label the elements of E in decreasing order:ε1 = log2(3/2) > ε2 = log3(4/3) > ε3 = log2(7/6) > . . .
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The structure of the set of CA numbers cotd.
(a) if ε /∈ E, σ(n)/n1+ε has a unique maximum attained at thenumber nε with prime exponents given by
ap(ε) =⌊logp
(p1+ε−1pε−1
)⌋−1
(b) if ε satisfies εi+1 < ε < εi for i = 1, 2, 3, . . . , then nε isconstant and we call it Ni. We have N1 = 2, N2 = 6, . . .
(c) if the sets Ep are pairwise disjoint (which is likely, but notcertainly known), then the set of CA numbers is equal to theset of Ni, i = 1, 2, 3, . . . . If this is the case, σ(n)/n1+ε attainsits maximum at the two points Ni and Ni+1
(d) if the sets Ep are not pairwise disjoint, then for eachεi ∈ Eq∩Ep, σ(n)/n1+εi attains its maximum at the four pointsNi, qNi, rNi and Ni+1 = qrNi
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Alaoglu & Erd ′′os [2]
F these authors have a slightly stronger definition:
F Defn: n is colossally abundant if
σ(n)/n1+ε > σ(k)/k1+ε for 1 6 k < n
σ(n)/n1+ε > σ(k)/k1+ε for k > n
F the effect of this is to make a unique choice from the 2 or 4possibilities in cases (c) and (d) above. But I will perform mycomputations with the Robin definition
F we will call these numbers strongly colossally abundant (SCA)if it is necessary to distinguish them from ordinary CA numbers
Keith Briggs RH and abundant numbers 8 of 22
Position of critical ε values
critical ε values
0 2 4 6 8 100
2
4
6
8
10
12
−log(ε)
p
The vertical lines mark thecritical ε values arising from thesmall primes on the y axis
Keith Briggs RH and abundant numbers 9 of 22
Density of critical ε values
strongly colossally abundant numbers
1 2 3 4 5 6 70
1
2
3
4
5
6
−log(ε)
log
log
(n)
The vertical lines markthe critical ε values
Keith Briggs RH and abundant numbers 10 of 22
Dependence of maximal prime on n
0 2 4 6 8 10 12 14100
101
102
103
104
105
106
loglog(n)
p max
This shows the maximalprime needed as a functionof log log (n)
Keith Briggs RH and abundant numbers 11 of 22
Method
F it is known that if RH is false, there will be a violation ofRobin's inequality which is a CA number . . .
F for a range of small ε > 0, I compute n by the A&E formula
F I compute RHS-LHS of Robin's inequality (let's call this the de-viation δ(n) ≡ eγ log log (n)−ρ(n)), and look for any violations(i.e. δ < 0)
F we can also plot η(n) ≡ eγ−ρ(n)/ log log (n)
F the following plots show the behaviour observed so far (toabout ε = exp(−25))
Keith Briggs RH and abundant numbers 12 of 22
Computational difficulties
F the exponents ap(ε) must be computed in interval arithmetic,to ensure they are correct and not corrupted by roundofferror. This means not just high precision arithmetic, butdynamically varying precision
F millions of primes are needed. Typically the n we deal withhave a huge tail of many primes to the power 1. It is fastestto precompute primes with a sieve, but then much storage isrequired.
F how do we vary ε to not miss any SCA numbers? (DONE)
F how to we compute explicit examples of WCA numbers?
F there are many other difficulties . . .
Keith Briggs RH and abundant numbers 13 of 22
Computational strategy
We keep a list z of records, containing: a prime p, log p, its exponent a, and acritical εc, which is the value of ε at which this exponent will next change (as ε isdecreased). We exclude 1 exponents, which are counted by ones. We first initialize:
. fix 0 < εstart 6 1. Then, for each prime p, compute a =⌊logp
(p1+εstart−1pεstart−1
)⌋−1
and store it in the z list if a > 2. If a = 1, just increment the variableones. Stop when a = 0. During this p loop, also update log(n) and ρ(n)
Then each step of the main loop consists of determining which of possible eventsA, B, or C occurs:
. A: a new prime (with exponent 1) is added, so we increment `ones'. Thishappens when εext = logp (1+p) is maximal, where p is the new prime
. B: the first prime with exponent 1 has its exponent raised to 2. This happenswhen εinc = logp ((p+1+1/p)/(p+1)) is maximal, where p is the prime inquestion
. C: a prime with exponent > 2 has its exponent incremented. This happenswhen εmax = logp ((1−pa+1)/(p−pa+1)) is maximal, where p is the prime inquestion and a its exponent
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Prime generation
For simple tests, just use a lookup list. With the BERNSTEIN option, crit eps14.cuses Bernstein’s quadratic sieve. Here we cannot just look up any prime; rather wehave a function to get the next prime and advance the internal state (also to peek atthe current prime without advancing). So the strategy is to use 2 prime generators- one (pg1) for the z list (which only grows slowly), and another (pg2) to see if thelist of 1s needs extending. pg2 goes a long way, but we rely on Bernstein’s code tokeep it efficient.
Keith Briggs RH and abundant numbers 15 of 22
Questions
F how does n depend on ε?
F how does the largest prime needed depend on ε?
F how does δ depend on ε?
F interesting observation: when p is large and ε is small (thesituation we are interested in), in the exponent formula ofAlaoglu and Erd ′′os it is already sufficient to use the first termof a Taylor expansion in ε, namely as ε → 0+,
logp
(p1+ε−1pε−1
)= logp
(p−1log p
)−logp(ε)+O(ε)
already has error less than 1/2, so the floor is the correctinteger. How can we exploit this?
F in the following plots the computed data is in red and hy-pothesized fits or trends are in blue
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Dependence of n on ε
strongly colossally abundant numbers
5 10 15 20 25 300
5
10
15
20
25
30
−log(ε)
log
log
(n( ε)
)
This shows that log log (n)at SCA numbers n ap-pears to be asymptoticallya linear function of − log ε.The line −1.779+0.947xis guesswork
Keith Briggs RH and abundant numbers 17 of 22
Density of CA numbers
0 2 4 6 8 100.0009
0.0010
0.0011
0.0012
0.0013
0.0014
log(n)/1010
CA
num
ber
dens
ity
Keith Briggs RH and abundant numbers 18 of 22
Dependence of δ on n
14 16 18 20 22 24 26 28−14
−13
−12
−11
−10
−9
−8
−7
loglog(n)
log
( δ)
This shows that log δ ap-pears to be asymptoticallya linear function of x =log log (n)
Keith Briggs RH and abundant numbers 19 of 22
Dependence of δ on n
26.969 26.970 26.971 26.972 26.973 26.974 26.975−13.1495
−13.1490
−13.1485
−13.1480
−13.1475
−13.1470
−13.1465
−13.1460
loglog(n)
log
( δ)
This is the last 100000values of the previousplot
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Deviation of log δ from a best-fit line
15 20 25−0.04
−0.02
0.00
0.02
0.04
0.06
0.08
loglog(n)
log
( δ)−
(0.3
23
33
6−
log
log
(n)/
2)
This shows the differencebetween log δ and a con-jectured best-fit line a−x/2, where x ≡ log log (n)
Keith Briggs RH and abundant numbers 21 of 22
References
[1] J. C. Lagarias An elementary problem equivalent to the Rie-mann hypothesis, Amer. Math. Monthly, 109 (2002), 534-543www.math.lsa.umich.edu/∼lagarias/doc/elementaryrh.ps
[2] L. Alaoglu & P. Erd ′′os On highly composite and similar num-bers, Trans. Amer. Math. Soc. 56 (1944) 448-469
[3] G. Robin Grandes valeurs de la fonction somme des diviseurset hypothèse de Riemann, J. Math. pures appl. 63 (1984)187-213
[4] S. Ramanujan Highly composite numbers. Annotated and witha foreword by J.-L. Nicholas and G. Robin, Ramanujan J. 1(1997) 119-153
[5] J.-L. Nicolas Ordre maximal d'un élément du groupe despermutations et highly composite numbers, Bull. Math. Soc Fr.97 (1969), 129-191
[6] P. Erd ′′os and J.-L. Nicolas Répartition des nombres superabon-dants, Bull. Math. Soc Fr. 103 (1975), 65-90
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