A relationship between certain sums over trivial and nontrivial zeros of the Riemann zeta-function

Zeta functions for the Riemann zeros

“…(79) below) arising as an instance of a generalized Poisson summation formula. Later, Delsarte introduced that function again (as φ(s) in [12]) to describe its poles qualitatively, displaying (only) its principal polar part at s = 1, as (2π) −1 /(s −1) 2 ; Kurokawa [24] made the same study at v = 1 4 , not only for ζ(s) but also for Dedekind zeta functions and Selberg zeta functions for PSL 2 (Z) [or congruence subgroups] (then, Zeta functions like (2) occur within the parabolic components); and Matiyasevich [27] discussed the special values θ n ≡ 2 Z (n, 1 4 ) (n ∈ N * ). Extensions in the style of the Lerch zeta function have also been studied ( [16], [23] chap.VI).…”

“…Indeed, a few symmetric functions over the Riemann zeros that resemble spectral functions have been well described, mainly I ( [10], [18], [23] chap.II). Zeta functions like (2) have also been considered, but almost solely to establish their meromorphic continuation to the whole a-plane -apart from the earliest occurrence we found: a mention by Guinand [17] (see also [8]) of the series (-Z(sl2, v = 0)) on one side of a functional relation (equation (79) (2) occur within the parabolic components) ; and Matiyasevitch [27] discussed the special values 9n * 2 Z (n, -1 ), (n E N*). Extensions in the style of the Lerch zeta function have also been studied ( [16], [23] chap.VI).…”

“…As for the values Z(m) themselves, they are given by equation (27) Table 1), as stated earlier by Matiyasevich [27]. (More recently, Z(I) and Z(2) also got revived in studies of the distribution of primes [33], [25]).…”

Yet another representation for reciprocals of the nontrivial zeros of the riemann zeta function

“…where {q} is the fractional part of q, so the positivity of the integrand is evident. From [7] γ − ln(4π)…”

Zeta Functions Over Zeros of General Zeta and L-Functions

“…Remark: the pair of mutually inverse relations (72) and (73) are clearly similar to the identities (53) and have the same origin. They extend to all primary functions L and all vvalues previous results written only for the Riemann case and v = 1 4 [20,27]. The resulting values of Z(σ, v) for general v are listed in Table 2 (lower half).…”