Zeros of the alternating zeta function on the line R(s)=1
Abstract
The alternating zeta function zeta*(s) = 1  2^{s} + 3^{s}  ... is related to the Riemann zeta function by the identity (12^{1s})zeta(s) = zeta*(s). We deduce the vanishing of zeta*(s) at each nonreal zero of the factor 12^{1s} without using the identity. Instead, we use a formula connecting the partial sums of the series for zeta*(s) to Riemann sums for the integral of x^{s} from x=1 to x=2. We relate the proof to our earlier paper "The Riemann Hypothesis, simple zeros, and the asymptotic convergence degree of improper Riemann sums," Proc. Amer. Math. Soc. 126 (1998) 13111314.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2002
 DOI:
 10.48550/arXiv.math/0209393
 arXiv:
 arXiv:math/0209393
 Bibcode:
 2002math......9393S
 Keywords:

 Mathematics  Number Theory;
 11M26
 EPrint:
 Typo corrected after equation (4)