Zeta functions for the Riemann zeros

Abstract

A family of Zeta functions built as Dirichlet series over the Riemann zeros are shown to have meromorphic extensions in the whole complex plane, for which numerous analytical features (the polar structure, plus countably many special values) are explicitly displayed.

Sharpenings of Li's Criterion for the Riemann Hypothesis

“…It was already known that λ n = n j=1 (−1) j+1 n j Z j [12, Equation (27)], and that the Z j in turn are complicated polynomials in the Stieltjes constants {γ k } k<j [22] (for λ n and γ k see also [21,3,4] and references therein). Now the latter relations boil down to [26,Equation (46)] simply by promoting logarithmic coefficients η j [11,2] (cf. also [16,Equation (12)…”

“…which extends to a meromorphic function in C having all its poles at the negative halfintegers, plus one pole at σ = + 1 2 [13] of polar part [26] Z( 1 2 + ε) = R −2 ε −2 + R −1 ε −1 + O(1) ε→0 ,…”

“…Other sums related to the Z(k) are Z j = ρ ρ −j [22,16,26] (often denoted σ j , but here we use σ as variable). It was already known that λ n = n j=1 (−1) j+1 n j Z j [12, Equation (27)], and that the Z j in turn are complicated polynomials in the Stieltjes constants {γ k } k<j [22] (for λ n and γ k see also [21,3,4] and references therein).…”

“…Remarks: -the η j are the Stieltjes [constants'] cumulants, up to some relabelings [26,27]; -the η j admit an arithmetic expression over the primes [2, Equation (4.1)]; see also [10], which cites [25] for the case η 0 = −γ; -relation (9) can be inverted also in closed form, by the same technique as for [26, Equation (48)]:…”

“…We will use the completed zeta function Ξ(s) (normalized as Ξ(0) = Ξ(1) = 1) and a symmetrized form of its Hadamard product formula [7,26],…”