B. Nyman, "A general prime number theorem", Acta Math. 81 (1949) 299-307.

This is a development of Beurling's generalised prime number theory [Acta Math. 68, 255-291 (1937)].

The "primes" are the members yn of a given real sequence (1 < y1 < y2 < ... yn , and the "numbers" are all possible products xn of them (1 < x1 <= x2 <= ...) with repetitions for multiple representations. Let be the number of yn <= x, N(x) the number of xn <= x, and = 1 + x1-s + x2-s + ... (s = + it, > 1).

It is asserted that the following three statements are equivalent:

(A) N(x) = ax + O(x/log-nx) as x, for some fixed a > 0 and every fixed n > 0
(B) |(n)(s)| < A|t|e ||-1 < A|t|e ( > 1, |t| >= e), for every e > 0, every integer n >= 0, and some A = A(e,n)
(C) = Li(x) + O(x/log nx) as x, for every fixed n > 0.

The argument is based on the scheme of implications: (1) A B, (2) B C, (3) C B, (4) B A. Of these (1) and (3) are established by partial integration and, in case (1), Hadamard's classical deduction from the inequality |3()4( + it) ( + 2it)| >= 1 ( > 1); while (2) and (4) are based on Parseval's formula. By the use of derivatives of high order the operations are confined essentially to the open half-plane 1.

[The argument does not seem to correspond logically to the enunciation, for under the headings B C and B A the author proves only (A and B) C and (C and B) A, respectively. Indeed, B can hardly be equivalent by itself to A or C, since it does not specify the behaviour of near s = 1. Thus, if yn = 2n, B is true but A and C are false. The main equivalence A C is, however, valid.]

Reviewed by A. E. Ingham



Beurling notes
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