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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