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

Source 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, $ΣR_T(x):=\sum_{0 1$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$\sum_ρR(x^ρ)$.