THE PRIME NUMBER THEOREM, AND ITS PROCFS*

Robert Breusch

(received Ik May, 1970)

1. The prime number theorem 'p.n.t.* concerns itself with the function* , defined for all positive x by

x(x) = number of primes less than or equal to x.

Thus *(10) = 4 because there are four prime numbers < 10, namely2,3,5, and 7 •

Obviously, *(x) < x (not all the positive integers < x are prime,e.g. 1 is not). The only additional thing knovn about «(x) by the endof the 18th century was that Urn *(x) - a fact that had been

x-* •proved by Euclid more than 2000 years earlier*

In the last decade of the 18th century, Gauss and Legendre seem to have been the first ones who conjectured, independently from each

other, that *(x) grows like arrived at this conjecture

probably by a close study of tables of prise numbers.

The p.n.t. asserts that Gauss and Legendre were right, and thatin fact

lim --- = 1X-* oo x/log X

We shall express this equation also by the statements: **(x) is

asymptotically equal to Yo"g"x* * an<* wr^ e ^ ^orm

*(x) ~log x

log x J 0 log t

x

It is easy to show that ~ J y . Thus

dtLog t

* Invited address delivered at the Fifth Hew Zealand MathematicsColloquium, held at Palmerston North, 13-15 May, 1970.

Math. Chronicle (19T0), 61-70.

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is another version of the p.n.t. This version is better in the

f x dtfollowing sense: for sufficiently large x, j-0g x < J ^

and proofs of the p.n.t. show that for all sufficiently large x,

x f x dt*(x) > » ;hence *(x) is closer to J y^'g ^ than to

xLog x

The function * has no simple relationship with any of the familiar functions of analysis. Therefore all the proofs make use of another function, ♦, which forms a bridge between * and some well- known functions of analysis, t is defined as follows:

for all positive n, let

M n ) = {log p if n = Pr •l0 otherwise

and for all positive x, let

+(*) = n§ x M n ) .

(Throughout this talk, the letter p will be reserved for prime numbers). Thus

♦(10) = log 2 + log 3 + log 2 + log 5 ♦ log 7 + log 2 + log 3 •

It can now be shown that the p.n.t. is equivalent with

*(x) - x ,

and that any formula of the form

♦(x) - x = 0(g(x))

implies that

.... 0 ( ^ 1 , , . ( . 1 - .

Here, and in the following, use has been made of the 0 symbol, first introduced by Landau (if I am correct), and defined as follows:

If g(x) > 0 for all sufficiently large x, then f(x) = 0(g(x)) implies the existence of positive xQ and C such that for all

x > xQ, |f(x) | < C*g(x) .

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2. There have been at least three different approaches to the p.n.t.

The first of these is due to Tchebychef, In a paper published in 1851. He was unable to prove the p.n.t,, but he obtained some impor­tant preliminary results which later paved the way for actual proofs. For positive x, Tchebychef conilders [x]! , where [ ] is the greatest integer function, [x]! contains as factors all the primes < x. A particular prime p will be contained In every multiple of p; there

are such multiples < x . p will be contained once more in

every multiple of p 2 : T— 1 of these are < x .U?2J *” r ,

Continuing in this way, we see that the multiplicity of p in [xj! is

[1 ] * (Kl4 [js] *..... Thi“ .. n[x]! = ** p

P< x

We take the natural logarithm of both sides, and call the left-hand expression T(x).

T(x) = log[x]! = n| x log n - £ x log p( | + j j j ] + ...

- £ x * “> [ i ] •

The last sum is also equal to x ♦ (■£)

Hameiy x +(£) = k| x * x M n ) .

~ n5 k

In this sum, a particular A(n) appears as often as there are positive

integers k with n < £ ; there are f^j numbers k of this kind. Thus

k§ x « S > = n< x T W *

Stirling's formula, we know that

T(x) = x log x - x + O(log x) = x log x + 0(x). Thus

(A) k| x +(£) = x log x + 0(x) .

This formula is an Immediate consequence of the p.n.t. (with a

sufficiently small remainder term): if t(x) ~ x, then

k§x ~ k i* . £ = x - k | x i = x 108 * + 0(x) •

Tchebychef's efforts, conversely to derive the p.n.t. from this formula, were unsuccessful. But he could prove that for all large x,

0.69x < *(x) < 1.39x .

This implies that, for large x,

O .69 t—--- < *(x) < 1.39 --- ,log x ' log x

thus proving that Gauss and Legendre had conjectured the correct rate of growth for *(x). Later he was able to narrow the gap and showthat^ for large x,

0.92x < *(x) < 1 .11x .

An easily obtained consequence of the relation

n i x * » >

is the formula

= x log x + 0(x)

(B) n < x ^ = log x + 0(1 ) ,

which will be used later on.

A few years after Tchebychef, Riemann attacked the problem from an entirely different direction. In a paper published in 1859, he starts out from the zeta-function which is defined for Re(s) > 1 by

?(s) = n=1 “5

(it is customary in number theory to call a complex variable s * 0 + it rather than z = x + iy, presumably because the letter z Is pre-empted as the argument in functions such as % and if •)£ is known to be regular in the whole plane, with exception of a simple pole of residue 1 at s = 1 . Por a > 1, £(s) can also be written in the form

£(s) = 11 (l + + ••• )>P P P 2®

where the product is taken over all the primes. Indeed, expanding

6k

this product, ve obtain a tun of terms , where n takes each possible _ n

product of prints,H p , exactly once. Bach aua unier the product sign is a geometric series* Thus

5(») n —p 1

P

Logarithmic differentiation leads to

E s08 P . - - S 1<W p(£i ♦ - 5 + ... )51 ' * P (1 -■*») p P P»“

P

" " nil ^

Of course, all these operations Involving infinite sums and infinite products require proofs of their legitimacy, and this is equally true for what follows. If a > 1, we form next, for x > 0,

a+i«* a+i«*

Assuming interchangeability of Integration and summation, we findthat

a+i« a+ioo

J r W d8 = * n£i A(n) 7 i(l)S ds •a-i» a-i»

By contour integration, it can now be shown, as a consequence of the residue theorem, that for positive b,

a+i»To if 0 < b < 1 2*1 if b > 1

a+loo

Ia-ioo

It follows for all non-integral x that

a+ioo

I T n| x 4 „ ) - - 2"1 * , ) .

a-i»

and therefore

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(®) *M = m f f"TT̂ds’ for a > '•a-i»

Riemann thus obtained an analytic expression for t(x); a sufficient amount of information about the zeta fuction might now lead to a proof of the p.n.t. Riemann himself was unable to complete the proof, but Hadamard and de la Vallee-Poussin succeeded in 1896, independent from each other. Starting out from formulas similar to the one above, and making use of properties of the zeta-function, they were the first ones to give complete proofs of the p.n.t.

In the year 19^8, Selberg and ErdSs surprised the mathematical world with a new proof of the p.n.t. which made no use whatever of the zeta function, or of any other tools of complex analysis. In this sense, their proof was •elementary' (which does not mean: easy). The proof was based on the following intriguing formula, first stated and proved by Selberg in a paper published in 19^7:

log x.^(x) + n^.x A(n)\|r (2) = 2x log x + 0(x) .

Like Tchebychef*s formula (A), this formula follows from the p.n.t. with a sufficiently small remainder term: if i|r(x) ~ x, then

log x-*(x) + n| x A(n)i|r(jj) ~ log x.x + n^ x A(n) 2

= x • (log x + £ ^lll) n< x n

= 2x log x + 0(x), by Tchebychef's formula (B).

But, like Tchebychef a hundred years earlier, Selberg was able to derive his formula directly. In fact, this particular formula has since been found to be truly elementary, in the sense that its proof requires very little preliminary knowledge. The difference between the two formulas is the following: in the earlier^formula, i|r(x) con­tributes only a small part of the total sum £ \Kr) >

k < x K

less than for x > xQ . Thus t(x) could deviate considerably from

its hoped-for value x, with the deviation being easily absorbed by small counter-deviations in ♦ (■£), f (^), ... .

By contrast, in the Selberg formula, t(x) contributes about half of the whole sum. Thus, if for some x, t(x) were, for example, much larger

than x, then most of the t(u) Vith A (n) / 0 would have to be correspon­dingly smaller. This in turn can be shown to imply that most of the

a+i«

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^(— ) (not only those with A(n) 4 0) would have to be less than their^ X X

expected value — : thus £ \K— ) would turn out to be significantly n n< x n

less than x.log x, and we would arrive at a contradiction to formula(A).

The details of such a proof are surprisingly complicated. So far, no easy proof of the p.n.t. has been found.

3. Right from the beginning, the proofs of the p.n.t. tried to include an estimate of the 'remainder term* i|r(x) - x. tet us look again at Riemann*s formula, formula (C). We integrate

over a rectangle with vertices

a - iW, a+ iN, -M + iN, -M - iN ,

with large positive numbers M and N chosen judiciously so as to avoid the poles of the integrand. Then it can be shown that as M and N tend to +00, the integrals over the three sides to the left of a = a, tend to zero. Thus we find by the residue theorem that

a+i»

*(x> ■ - d a ds = - 2 resldues>

where the sum is taken over all the poles of the integrand to the left of a = a. Poles are at s = 0, at s = 1 (where £ has a pole), and at s = P if P is any one of the zeros of £. The residues at these poles can be found easily. It follows that

!2l. i i f .\jf(x) = X - - Zn ' M o J p p

Thus knowledge about the zeros of £ becomes vitally important.Riemann knew the following facts about the zeros of £ : £ has simple zeros at the negative even integers; they are usually referred to as the 'trival zeros' • All the other zeros p of 5 are such that0 < Re(p) < 1. There are infinitely many of these 'non-trivial zeros1. They are distributed symmetrically with respect to the real axis, and also symmetrically with respect to the line a = ^ .Riemann made the famous conjecture that all the non-trivial zeros of £ are on the line o = ^ . So far, after more than a hundred years, his conjecture has been neither proved nor disproved.

The following can now be shown to be true. If there exists a positive a- such that Re(p) < 1 - a for all zeros p of £, then

V(x) - x = 0(xlwaf€), for every € > 0 ;

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and, conversely, if i|r(x) - x = Ofc1”0**) for every 6 > 0, then S(s) ^ 0 for cr > l - a .

In particular,

♦(x) - x = 0( x ^ €)

if and only if Riemann*s conjecture is true. In order to facilitate a comparison vith results that were actually achieved, we write the remainder in the form

*(x) - x = 0(x*e"g^ ) .

Thus g(x)^> (cuc)log x if and only if £(s) / o for a > 1 - a . de la Vallee-Poussin proved in 1899 that g(x) > c.Aog x, and

Vinogradov improved this in 1958 to g(x) > (log x ) ^ 5”€ . When the first elementary proofs were published, hope ran high that they might perhaps lead to significantly smaller remainder terms, maybe of the

i -aform x . A s stated before, this in turn would lead to information about the zeros of £ , so that perhaps Riemann*s conjecture could be attacked via the back door, so to speak. This hope has turned out to be unfounded, so far anyway. In fact, the best estimates obtained by the elementary methods are not as good as those achieved previously. They show that

t(x) - x = o ( ---;vLog x) '

thus g(x) > 0 log log x, where 3 has increased spectacularly within a decade. Around 1955* Van der Corput showed that 3 > — . This

1 1 3 “ 200was increased in succession to — ; — ; — : 1 ; ’arbitrarily large* . The

10 6 4last result is due to Bombleri (1962) and Wlrslng (196k). Here then is a summary of the results:

t(x) - x = 0(x«e”g^ ) with

g(x) > (a - c)log x if £(s) ^ 0 for a > 1 - a

g(x) > (log x) by analytic methods

g(x) > 3 * log log x for all 3, by elementary methods.

Even for large 3,

148 — = 0 .* (log x)a/s_s

Therefore, the best result achieved by elementary methods is not as good as Vinogradov's estimate. In turn, for every positive a (and

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sufficiently small € , of course)

=°(loe x)-

X -*00

Therefore the best result known is inferior to what would be true if we knew that 5(s) ^ 0 for o"5 0.99 > for instance. Thus there is room for improvement on all fronts.

Let me conclude this talk with a bit of day-dreaming. Tchebychef*s

formula (B) says that £ - log x is bounded; it does not

say that this difference converges. However, the p.n.t. with a sufficiently small remainder term implies that indeed

lim ( £ i M . log x) exists. _ vn< x n B 'x -k» _

Conversely, the existence of this limit implies the p.n.t. Now it is found that this limit exists, and is -7 , where 7 is Euler*s constant,defined by

7 = 108 x > • x-*« —

This is a rather extraordinary coincidence, and one feels that there ought to be some good and simple reason for it. If such a reason could be found, involving a small enough remainder term for

n< x ”"n"^ " log x + ^ in turn vould provide a similarly

small remainder term for y(x) - x, and it might lead to information about the zeros of £. This in extremely unlikely, of course, but then, who knows?

Books on prime numbers

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Leipzig, 1909- 2nd ed., Chelsea, New York, 1953-

E. Landau, Vorlesungen liber Zahlentheorie, Leipzig, 1927. Chelsea, New York, 1955.

T. Estermann, Introduction to modern prime number theory, Cambridge Tracts in Mathematics and Mathematical Physics. U1, Cambridge University Press, London, 1952.

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G.H. Hardy and E.M. Wright, An introduction to the theory of numbers. Clarendon Press, Oxford, 1938. ^th ed., i960.

K. Prachar, Primzahlverteilung, Die Grundlehren der mathematischen Wissenschaften. 91, Springer-Verlag, Berlin, 195T•

An excellent summary is found in

Wolfgang Schwarz, Der Primzahlsatz, Ueberblicke Mathematik 1 (1968), 3 5 - 6 1 .

Amherst College, Amherst, Mass. University of Waikato.