Volume no :5, Issue no: 1, March (2011)

ON TWO CONTROVERSIAL ESTIMATES IN NUMBER THEORY

Author's: Y.-F. S. Pétermann
Pages: [1] - [9]
Received Date: February 22, 2011
Submitted by: Dae San Kim

Abstract

An unsubstantiated 1958 claim of Vinogradov [15] and Korobov [8] concerning the remainder term in the prime number theorem, never clearly withdrawn by its authors, has been widely reproduced in the literature since then and still occasionally appears even in serious publications.

Korobov’s paper [8] is also linked with another less familiar controversy regarding two asymptotic estimates of a sum involving the Euler due respectively to Walfisz [19] and Saltykov [13].

My two purposes here are to offer some comments on the first issue, based on bibliographic references, and to settle the second one by pointing out a mistake in [13]. This note is largely based on a talk I gave a few years ago at seminars and at a conference.

Keywords

Euler’s function’s average, Weyl’s exponential sums, prime number theorem, zeros of the zeta function.