A famous and almost ubiquitous formula, going back to Riemann and von Mangoldt, expresses the prime-counting function \(\pi(x)\) in terms of the zeros \(\rho\) of the Riemann zeta-function. It is often encapsulated in the memorable formula \(\pi(x)=R(x)-\sum_{\rho}R(x^\rho)\), where \(R(y)\) denotes Riemann’s \(R\)-function. In this talk, we will present our recent proof that this expression, which has appeared throughout the literature for more than fifty years, is not in fact a valid mathematical identity. Rather, it turns out to be the product of a surprisingly persistent -- though ultimately unjustified -- optimism concerning the convergence and interpretation of the sum over the zeros.
Latest news about the prime-counting function \(\pi(x)\)
06.10.2026 13:15 - 14:45
Organiser:
H. Grobner, A. Mellit, B. Szendroi
Location: