An improved bound for the de Bruijn-Newman constant
Andrew Odlyzko
AT&T Labs - Research
amo@research.att.com
The Riemann Hypothesis is equivalent to the conjecture that the
de Bruijn-Newman constant LAMBDA satisfies LAMBDA <= 0. However,
so far all the bounds that have been proved for LAMBDA go in the
other direction, and provide support for the conjecture of Charles
Newman that LAMBDA >= 0. This paper shows how to improve previous
lower bounds and prove that
-2.7 / 10^9 < LAMBDA.
This can be done using a pair of zeros of the Riemann zeta function
near zero number 10^{20} that are unusually close together. The new
bound provides yet more evidence that the Riemann Hypothesis, if true,
is just barely true.