The generalized divisor problem and the Riemann hypothesis

Abstract

Let dz(n) be a multiplicative function defined bywhere s = σ + it, z is a. complex number, and ζ(s) is the Riemann zeta function. Here ζz(s) = exp(z log ζ(s)) and let log ζ(s) take real values for real s > 1. We note that if z is a natural number dz(n) coincides with the divisor function appearing in the Dirichlet-Piltz divisor problem, and d-1(n) with the Möbious function.

Abel-Tauber Process and Asymptotic Formulas

“…The former was generalized by Nakaya [38]. These four authors treated the proper main term in the form of a contour integral and were able to exhibit an error estimate as good as that for the PNT and then showed that it can be expanded into an asymptotic formula.…”

The moments of the zeta-function on the line σ = 1

“…where N > 1 is an arbitrary but fixed integer, | z \ < A for any fixed A > 0, .2) we shall use a recent result of H. Nakaya [5], who proved, uniformly for \z I < A and x > 3, A proof of this result, due essentially to I. M. Vinogradov, may be found in Ch. 6 of [3].…”

“…2) is only by a log-factor of a smaller order of magnitude than the main term, the use of (3.2) is insufficient to produce an approximate functional equation for ζ (1 + if) capable of proving (2.4). Instead of (3.2) we shall use a recent result of H. Nakaya [5], who proved, uniformly for \z I < A and x > 3, In £ we take r -1 /log X, so that the integrand of the last integral is regular in £ and on its boundary. In view of (3.5) it follows that the integral converges absolutely for σ > 1, which means that we have obtained analytic continuation of the left-hand side of (3.8) that is valid for σ = 1.…”