Erdos 1964 arithmetical tauberian theorems

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P. Erdős, A. E. Ingham: Arithmetical Tauberian theorems, Acta Arith. 9 (1964), 341--356 (MR 31 #1228; Zentralblatt 127,271).

Erdős and Ingham replace the sequence of all positive integers, in the classical Tauberian setting behind the prime number theorem, by a finite or infinite real sequence 1 < a_1 <= a_2 <= ... with A = sum 1/a_n finite (printed p.341 / physical p.1), and ask when the hypothesis f(x) + sum_n f(x/a_n) = (1 + A + o(1))x implies f(x) = (1+o(1))x. An Abelian averaging lemma gives Theorem 1: the implication holds for all bounded-on-bounded-intervals f when A < 1, while for A = 1 and real f one only gets limsup and liminf of f(x)/x equal to 1 plus or minus c (0 <= c <= infinity). Theorem 2 shows that for A = 1 and f in class I the conclusion holds unless all a_n are odd powers of a single alpha > 1, and Theorem 3 handles special configurations with A > 1 by elementary dilation subtraction. The central Theorem 4 (printed p.347 / physical p.7), proved with Wiener's general Tauberian theory, states that the implication holds for all f in class I if and only if Z(s) = 1 + sum a_n^{-s} does not vanish on the line Re s = 1. Class I (p.342 / physical p.2) consists of nonnegative nondecreasing locally bounded real functions, equal to zero for x < 1. The initial sequence consists of real numbers and need not be strictly increasing or integral.

For Problem 967 the distinct-integer question occurs later, on printed p.355 / physical p.15. There the authors report no example with distinct integer a_n for which Z(s) vanishes on Re s = 1, and identify {2,3,5} as a simple undecided case. They do not call it the simplest case. The modern strictly increasing integer formulation is Question 1.1 in Yip's 2025 preprint, p.1. These historical statements do not establish a current status by themselves. The full Tauberian proof has not been reconstructed or independently reviewed here; the other result summaries below retain their earlier unreviewed scope.

Source: https://users.renyi.hu/~p_erdos/1964-05.pdf. No notice is printed in the file's text layer; the publisher's article page offers the PDF as a "Free download under CC-BY license", a Creative Commons Attribution license with no version or URL named (https://www.impan.pl/get/doi/10.4064/aa-9-4-341-356, read 2026-10-02).

Bears on. #967

Results to transcribe.

• Lemma (Abelian principle): For real g bounded on bounded intervals (class R) and g*(x) = sum g(x/a_n), the lower and upper densities satisfy Aliminf(g/x) <= liminf(g/x) <= limsup(g*/x) <= A*limsup(g/x).
• Theorem 1: If A < 1 the hypothesis implies f(x) = (1+o(1))x for all f in the basic class; if A = 1 and f is real it only gives limsup and liminf of f(x)/x equal to 1 plus or minus c, with 0 <= c <= infinity.
• Theorem 2: For A = 1 and f in class I the conclusion holds unless a_n = alpha^{r_n} for a fixed alpha > 1 and odd integers r_n, in which case it can fail.
• Theorem 3: For certain configurations with A > 1, where a subset S and its dilate satisfy a reciprocal-sum inequality, the implication still holds by elementary means.
• Theorem 4 (printed p.347 / physical p.7): For the initial finite or infinite real sequence 1 < a_1 <= a_2 <= ... with finite reciprocal sum, the Tauberian implication holds for every f in class I if and only if Z(s) = 1 + sum a_n^{-s} has no zero on Re s = 1. Class I is defined on p.342.
• The case {2,3,5} (printed p.355 / physical p.15): The authors cannot decide whether Z(s) vanishes on Re s = 1 for this distinct-integer sequence. It is described as a simple case beyond the reach of their theorems.