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 features have strangely many particular, but principally overlooked homes, that are surveyed the following in an available and artificial demeanour, after which compiled in several tables. No past booklet has addressed this overlooked subject in analytic quantity concept. Concretely, this instruction manual might help a person confronted with symmetric sums over zeros like Riemann’s. extra typically, it goals at reviving the curiosity of quantity theorists and complicated analysts towards these unusual services, at the 150th anniversary of Riemann’s work.

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In fact, ζ(x) does not vanish on the line Re x = 1 either. While this is a key step toward the prime number theorem, a major arithmetical result, both the fact itself and its derivation seem to lie off our main analytical track; so, we merely quote the result (proved in all books). This further restricts the Riemann zeros to the open critical strip 0 < Re x < 1. No zeros can lie on the real segment (0,1) either: indeed, ∞ (1 − 21−x ) ζ(x) = (−1)k−1 k −x converges for Re x > 0, k=1 and for all x > 0 this series is alternating and shows a strictly positive sum.

Here we recapitulate the facts gathered in this chapter that serve that purpose. 29) of the zeta-regularized form, Δ∞ (x) = exp −Z (0, x) , goes in the wrong direction for us: it precisely uses a zeta function that we aim at understanding (plus analytical continuation in totally abstract form). Fortunately, given an eligible sequence {xk } of order μ0 in the sense of Sect. 13), we now basically have a mechanical zeta-regularization rule. 47) μn ∈N / m=0 m (recalling that Hm = j=1 1/j are the harmonic numbers).

We follow the standard exposition, as condensed in [57, Sect. 4] for instance; usually, however, the bounds for [ζ /ζ](x) are intended to hold everywhere, whereas we focus on Im x 1, which simplifies some equations. The logic goes as follows: (1) bound ζ /ζ for Re x ≥ 2, and specially at the edge; (2) using this, bound the fluctuations in the distribution of the zeros’ ordinates {Im ρ}; (3) in turn, use the latter results to bound [ζ /ζ](x) when −1 < Re x < 2. We then develop those successive points in (partial) detail.

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