On a smoothed average of the number of Goldbach representations (Q6606728)

Let \N\[\N\psi_2(n)=\sum_{m+m'=n}\Lambda(m)\Lambda(m'), \N\]\Nwhere \(\Lambda\) is the von Mangoldt function. In the paper under review, the authors study a smoothed average for the number of Goldbach representations which are multiples of \(q\). This average is defined as\N\[\NF_q(N)=\sum_{\substack{n \\ q|n}}\psi_2(n)e^{-n/N},\N\]\Nand when \(q = 1\) we let \(F(N) := F_1(N)\) for averaging over all positive integers. In detail, the result is the following theorem.\N\NTheorem 1.1. Suppose the Generalized Riemann Hypothesis holds for the Dirichlet \(L\)-functions \(L(s, \chi)\) with characters \(\chi \pmod q\). Then for \(q\geq2\),\N\[\NF_q(N)=\frac{1}{\phi(q)}F(N)+O(N\log N\log q),\N\]\Nand when \(q=1\),\N\[\NF(N)=N^2-2\sum_{\rho}\Gamma(\rho)N^{\rho+1} + O(N),\N\]\Nwhere the sum runs over the non-trivial zeros of the Riemann zeta-function, and \(\Gamma(s)\) is the gamma function.\N\NFor the entire collection see [Zbl 1530.11003].