On divergence related to Riemann--von Mangoldt's explicit formula of the prime-counting function

Abstract

An explicit formula for the prime-counting function $π(x)$, usually attributed to Riemann and von Mangoldt, is prominently stated as the equation $π(x)=R(x)-\sum_ρR(x^ρ)$, where the sum runs over all zeros $ρ$ of the Riemann $ζ$-function, the non-trivial ones being ordered by increasing absolute value of their imaginary parts and counted with multiplicity. This particularly entails the claim that the partial sums over the non-trivial zeros, $ΣR_T(x):=\sum_{0<|\Im m(ρ)|\le T} R(x^ρ)$ converge as ${T\to\infty}$. Writing $Θ:=\sup\{\Re e(ρ):\ ζ(ρ)=0,\ 0<\Re e(ρ)<1\},$ for what has recently been called ``Riemann&#39;s constant&#39;&#39;, we prove that, for every fixed $x>1$ and every $θ<Θ$, the sums $ΣR_T(x)$ are not $O(T^θ)$. As a consequence, $\limsup_{T\to\infty}|ΣR_T(x)|=\infty$ and $\sum_ρR(x^ρ)$ diverges. We conclude the paper by showing that an adapted, but simpler strategy also gives the divergence of the contribution of the trivial zeros to $\sum_ρR(x^ρ)$.