Mathematika

Research Article

On Negative Moments of the Riemann Zeta-Function

S. M. Goneka1

a1 Department of Mathematics, University of Rochester, Rochester, NY 14627, U.S.A.

The purpose of this paper is to take some first steps the investigation of the negative moments

S0025579300013589_eqnU1

where k>0 and 12, and the related discrete moments

S0025579300013589_eqnU2

where runs over the complex zeros of the zeta-function. We assume the Riemann hypothesis (RH) throughout; it then follows that Ik(, T) converges for every k > 0 when > but for no k = when =. We further note that Jk(T) is only defined for all T if all the zeros are simple and, in that case, Ik(, T) converges for all k<.

(Received July 26 1988)

Key Words:

  • 10H05: NUMBER THEORY; Multiplicative Theory; Riemann's zeta-function