On the validity of the Euler product inside the critical strip

Abstract

The Euler product formula relates Dirichlet $L(s,\chi)$ functions to an infinite product over primes, and is known to be valid for $\Re (s) >1$, where it converges absolutely. We provide arguments that the formula is actually valid for $\Re (s) > 1/2$ in a specific sense. Namely, the logarithm of the Euler product, although formally divergent, is meaningful because it is Ces\`aro summable, and its Ces\`aro average converges to $\log L (s,\chi)$. Our argument relies on the prime number theorem, an Abel transform, and a central limit theorem for the Random Walk of the Primes, the series $\sum_{n=1}^N \cos\left(t\log p_n\right)$, and its generalization to other Dirichlet $L$-functions. The significance of ${\Re(s) > 1/2}$ arises from the $\sqrt{N}$ growth of this series, since it satisfies a central limit theorem. $L$-functions based on principal Dirichlet characters, such as the Riemann $\zeta$-function, are exceptional due to the pole at $s=1$, and require $\Im (s) \neq 0$ and a truncation of the Euler product. Compelling numerical evidence of this surprising result is presented, and some of its consequences are discussed.