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- W4301687510 abstract "Let $Lambdaleft(nright)$ be the von Mangoldt function and $r_{G}left(nright) := sum_{m_1 + m_2=n} Lambda left(m_1 right) Lambdaleft(m_2 right)$ be the counting function for the numbers that can be written as sum of two primes (that we will call Goldbach numbers, for brevity) and let $widetilde{S }left(zright) := sum_{ngeq1} Lambdaleft(nright) e^{-nz}$, with $zinmathbb{C}$, $mathrm{Re}left(zright)>0$. In this paper we will prove the identity $$widetilde{S}left(zright) = frac{e^{-2z}}{z}-sum_{rho}z^{-rho} Gamma left(rhoright) + sum_{rho} left(z^{-rho} gammaleft(rho,2zright) - frac{2^{rho}e^{-z}}{rho} right) + Gleft(zright)$$ where $gammaleft(rho,2zright)$ is the lower incomplete Gamma function, $rho=beta+igamma$ runs over the non-trivial zeros of the Riemann Zeta function and $Gleft(zright)$ is a sum of (explicitly calculate) elementary function and complex Exponential integrals. In addition we will prove that begin{align*} sum_{nleq N} r_G left(nright) left(N-nright) = & frac{N^{3}}{6} - 2sum_{rho}frac{left(N-2right)^{rho+2}}{rholeft(rho + 1right)left(rho+2right)} + & sum_{rho_1} sum_{rho_2} frac{Gammaleft(rho_{1}right) Gammaleft(rho_{2}right)} {Gammaleft(rho_{1} + rho_{2}+ 2right)} N^{rho_1 + rho_2+1} + Fleft(Nright) end{align*} where $N>4$ is a natural number and $Fleft(Nright)$ is a sum of (explicitly calculate) elementary functions, dilogarithms and sums over non-trivial zeros of the Riemann Zeta function involving the incomplete Beta function." @default.
- W4301687510 created "2022-10-05" @default.
- W4301687510 creator A5040885881 @default.
- W4301687510 date "2017-11-23" @default.
- W4301687510 modified "2023-09-23" @default.
- W4301687510 title "Some identities involving the Ces`aro average of Goldbach numbers" @default.
- W4301687510 doi "https://doi.org/10.48550/arxiv.1711.08610" @default.
- W4301687510 hasPublicationYear "2017" @default.
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