2015/09/30 by Guilherme França, André LeClair
Mathematics · Physics and Astronomy · #Analytic Number Theory Research #Cusp (singularity) #Dirichlet L-function #Dirichlet distribution #Dirichlet series #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Multiplicative function #Ramanujan's sum #Random walk #Riemann hypothesis #Series (stratigraphy) #Zero (linguistics) #math-ph #math.MP #math.NT
paper · pdf · doi:10.1142/s0219199717500857
published as Communications in Contemporary Mathematics (2017) 1750085 · Matches published version
openalex created_date 2016/06/24 · openalex publication_date 2017/10/10 · arxiv created 2017/11/15 · arxiv updated 2017/11/16 · openalex updated_date 2026/08/05
The aim of this paper is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of [Formula: see text]-functions, namely, non-principal Dirichlet and those based on cusp forms. The simplest example of the latter is based on the Ramanujan tau arithmetic function. For both classes, we prove that if a particular trigonometric series involving sums of multiplicative characters over primes is [Formula: see text], then the Euler product converges in the right half of the critical strip. When this result is combined with the functional equation, the non-trivial zeros are constrained to lie on the critical line. We argue that this [Formula: see text] growth is a consequence of the series behaving like a one-dimensional random walk. Based on these results, we obtain an equation which relates every individual non-trivial zero of the [Formula: see text]-function to a sum involving all the primes. Finally, we briefly mention important differences for principal Dirichlet [Formula: see text]-functions due to the existence of the pole at [Formula: see text], in which the Riemann [Formula: see text]-function is a particular case.