On the Prime Number Theorems

Abstract

A few years ago, Watson succeeded in proving an asymptotic congruence property formulated by Ramanujan, namely, the following theorem: If m and Ic are fixed positive integers, there are between n = 1 and n = N only o (N) integers n for which the sum of the (2m 1) -th powers of all divisors of n is not a multiple of k. This implies, in the particular case 2m 1 = 11, that Ramanujan's r(n) is divisible by 691 except for o(N) values of n. Actually, Watson has improved on o(N) by logarithmical factors (for arbitrary m, kc), and has even proved asymptotic formulae for certain summatory functions on which the o-estimates depend 1; asymptotic formulae established by the classical methods of the theory of primes (that is, by contour integrations, based on estimates of 4(s) in a certain domain containing the line a = 1 in its interior). The singularities of the generating Dirichlet series and, correspondingly, the proof of the asymptotic formulae are similar to those occurring in Landau's result,2 according to which the number of the positive integers which are less than x and are representable as a sum of two squares is asymptotically proportional to x/ (log x) '. The following considerations, which deal with the general analytical background of asymptotic laws of this type, seem to be warranted in view of Hardy's recent presentation 3 of the results just meiitioned. The asymptotic formula, given by Hardy in his discussion of Landau's problem, is false, the passage from the order of the singularity (at s = 1) to the order of the summatory function being erroneous. This can be seen, without any Tauberian argument, from Abelian reasons alone. (Among all the possible orders, Landau's exponent, 1, represents the only case in which Hardy's calculation becomes correct.) However, my quarrel is not with this lapsus calatmi (which does not occur in Watson's paper), but with the method of approach. In fact, Hardy refers (as Watson did) to the parallel problem of