Volume 1, pp. 104-111, 1993.

A new Lehmer pair of zeros and a new lower bound for the de Bruijn-Newman constant Λ

G. Csordas, A. M. Odlyzko, W. Smith, and R. S. Varga

Abstract

The de Bruijn-Newman constant $\mathrm{\Lambda }$ has been investigated extensively because the truth of the Riemann Hypothesis is equivalent to the assertion that $\mathrm{\Lambda }\le 0$. On the other hand, C. M. Newman conjectured that $\mathrm{\Lambda }\ge 0$. This paper improves previous lower bounds by showing that $$-5.895\cdot {10}^{-9}<\mathrm{\Lambda }.$$ This is done with the help of a spectacularly close pair of consecutive zeros of the Riemann zeta function.

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Key words

Lehmer pairs of zeros, de Bruijn-Newman constant, Riemann Hypothesis.

AMS subject classifications

30D10, 30D15, 65E05.