Zeta Functions over Zeros of Zeta Functions by André Voros PDF
By André Voros
The well-known zeros of the Riemann zeta functionality and its generalizations (L-functions, Dedekind and Selberg zeta features) are analyzed via a number of zeta features outfitted over these zeros. those ‘second-generation’ zeta services have unusually many particular, but mostly not noted homes, that are surveyed the following in an obtainable and artificial demeanour, after which compiled in several tables. No earlier ebook has addressed this overlooked subject in analytic quantity idea. Concretely, this instruction manual may also help a person confronted with symmetric sums over zeros like Riemann’s. extra regularly, it goals at reviving the curiosity of quantity theorists and intricate analysts towards these unexpected capabilities, at the 150th anniversary of Riemann’s work.
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Extra resources for Zeta Functions over Zeros of Zeta Functions
Explicit results can also be reached for the spectral zeta functions of this problem by methods parallel to those we will use for superzeta functions; those results are listed in Appendix B. 11) in similar ways. , [50, 76]. However, we wish to rather emphasize several diﬀerences.
10) was proved by Kurokawa  to be meromorphic in the whole σ-plane while Matiyasevich  studied the special values θm ≡ 2 Z∗ (m) (m = 1, 2, . ); Z∗ (1) and Z∗ (2) resurfaced lately in studies on the distribution of primes [93, pp. 13)]. A full-ﬂedged family arose still later, speciﬁcally Z (ﬁrst kind). (Earlier it came up only in exercises posed by Patterson [89, Ex. 3] for Re z > 1, followed by Schr¨ oter and Soul´e  for general z, described the meromorphic continuation in s of ξ(s, z) ≡ (2π)s Z (s | z − 12 ), but with focus on the parametric special value z → ξ (0, z).
11) in similar ways. , [50, 76]. However, we wish to rather emphasize several diﬀerences.
Zeta Functions over Zeros of Zeta Functions by André Voros