Explicit formula for the average of Goldbach and prime tuples representations

TL;DR

This paper derives explicit formulas for the averages of Goldbach and prime tuple counting functions using advanced special functions and the zeros of the Riemann Zeta function, providing asymptotic and truncated versions.

Contribution

It introduces explicit formulas for the averages of Goldbach and prime tuples, involving elementary functions, special functions, and non-trivial zeros of the Riemann Zeta function, with asymptotic and weighted analyses.

Findings

Explicit formulas involving special functions and zeta zeros.

Asymptotic and truncated formulas for average counts.

Observations on Cesàro weighted averages.

Abstract

Let $\Lambda\left(n\right)$ be the Von Mangoldt function, let \[ r_{G}\left(n\right)=\underset{{\scriptstyle m_{1}+m_{2}=n}}{\sum_{m_{1},m_{2}\leq n}}\Lambda\left(m_{1}\right)\Lambda\left(m_{2}\right), \] \[ r_{PT}\left(N,h\right)=\sum_{n=0}^{N}\Lambda\left(n\right)\Lambda\left(n+h\right),\,h\in\mathbb{N} \] be the counting function of the Goldbach numbers and the counting function of the prime tuples, respectively. Let $N>2$ be an integer. We will find the explicit formulae for the averages of $r_{G}\left(n\right)$ and $r_{PT}\left(N,h\right)$ in terms of elementary functions, the incomplete Beta function $B_{z}\left(a,b\right)$, series over $\rho$ that, with or without subscript, runs over the non-trivial zeros of the Riemann Zeta function and the Dilogarithm function. We will also prove the explicit formulae in an asymptotic form and a truncated formula for the average of $r_{G}\left(n\right)$. Some observation about these formulae and the average with Ces\`aro weight \[ \frac{1}{\Gamma\left(k+1\right)}\sum_{n\leq N}r_{G}\left(n\right)\left(N-n\right)^{k},\,k>0 \] are included.