2014/10/13 by França, Guilherme, LeClair, André
#FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1410.3520
The Euler product formula relates Dirichlet L(s,χ) 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 Cesàro summable, and its Cesàro average converges to log L (s,χ). 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 ∑n=1N cos(tlog pn), and its generalization to other Dirichlet L-functions. The significance of \Re(s) > 1/2 arises from the √(N) growth of this series, since it satisfies a central limit theorem. L-functions based on principal Dirichlet characters, such as the Riemann ζ-function, are exceptional due to the pole at s=1, and require \Im (s) ≠ 0 and a truncation of the Euler product. Compelling numerical evidence of this surprising result is presented, and some of its consequences are discussed.