Linear relations connecting the imaginary parts of the zeros of the zeta function. (English) Zbl 0216.03601

Bateman, P. T.; Brown, J. W.; Hall, R. S.; Kloss, K. E.; Stemmler, R. M.

Computers in Number Theory, Proc. Atlas Sympos. No. 2, Oxford 1969, 11-19 (1971).

Summary: Let \(\gamma_1, \gamma_2, \ldots\) be the imaginary parts of the zeros of the Riemann zeta-function in the upper half-plane, arranged in order of magnitude. This paper considers sums of the form \[ \sum_{n=1}^N c_n\gamma_n, \tag{*} \] where each \(c_n\) has one of the five values \(-2,-1,0,1,2\), not all the \(c_n\) are zero, and at most one \(\vert c_n\vert\)has the value \(2\). In the theory of the distribution of primes, it is important to show whether or not any of the sums (*) are zero. In particular, if at most a finite number of these sums are zero, then we know that \(x^{-\frac12} \sum_{n\le x} \mu(n)\) is unbounded both above and below.
In this paper that sum of the form (*) which is closest to zero is exhibited for each value of \(N\) from 1 to 20. This minimal absolute value is compared with the value predicted from crude probability considerations. The agreement is so close as to suggest that one would not be likely to find a zero sum of the form (*) within a finite amount of computer time.
[For the entire collection see Zbl 0214.00106.]

Reviewer: P. T. Bateman

MSC:

11M26

Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses

Keywords:

imaginary parts; zeros of Riemann zeta-function; upper half-plane

Citations:

Zbl 0214.00106