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

Abstract

An explicit formula for the prime-counting function $\pi (x)$, usually attributed to Riemann and von Mangoldt, is prominently stated as the equation $\pi (x)=R(x)-{\sum }_{\rho }R(x^{\rho })$, where the sum runs over all zeros $\rho$ of the Riemann $\zeta$-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, $\Sigma R_T(x):={\sum }_{0<|\Im m(\rho )|\le T}R(x^{\rho })$ converge as $T\to \infty$. Writing $\Theta :=\sup \{\Re e(\rho ):\zeta (\rho )=0,0<\Re e(\rho )<1\},$ for what has recently been called ``Riemann's constant'', we prove that, for every fixed $x>1$ and every $\theta <\Theta$, the sums $\Sigma R_T(x)$ are not $O(T^{\theta })$. As a consequence, ${\mathrm{limsup}}_{T\to \infty }|\Sigma R_T(x)|=\infty$ and ${\sum }_{\rho }R(x^{\rho })$ 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 ${\sum }_{\rho }R(x^{\rho })$.