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Zeta functions over zeros of zeta functions and an exponential-asymptotic view of the Riemann hypothesis. (English) Zbl 1365.11107

Summary: We review generalized zeta functions built over the Riemann zeros (in short: ‘superzeta’ functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz zeta function. As a concrete application, a superzeta function enters an integral representation for the Keiper-Li coefficients, whose large-order behavior thereby becomes computable by the method of steepest descents; then the dominant saddle-point entirely depends on the Riemann hypothesis being true or not, and the outcome is a sharp exponential-asymptotic criterion for the Riemann hypothesis that only refers to the large-order Keiper-Li coefficients. As a new result, that criterion, then Li’s criterion, are transposed to a novel sequence of Riemann-zeta expansion coefficients based at the point 1/2 (vs 1 for Keiper-Li).

MSC:

11M41 Other Dirichlet series and zeta functions
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11M35 Hurwitz and Lerch zeta functions
30B40 Analytic continuation of functions of one complex variable
30E15 Asymptotic representations in the complex plane
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
11M06 \(\zeta (s)\) and \(L(s, \chi)\)

Biographic References:

Aoki, Takashi
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