2013/09/26 by Guilherme França, França, Guilherme, André LeClair +1 · 3 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1309.7019
openalex publication_date 2013/09/26 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
We generalize our recent construction of the zeros of the Riemann ζ-function to two infinite classes of L-functions, Dirichlet L-functions and those based on level one modular forms. More specifically, we show that there are an infinite number of zeros on the critical line which are in one-to-one correspondence with the zeros of the cosine function, and thus enumerated by an integer n. We obtain an exact equation on the critical line that determines the n-th zero of these L-functions. We show that the counting formula on the critical line derived from such an equation agrees with the known counting formula on the entire critical strip. We provide numerical evidence supporting our statements, by computing numerical solutions of this equation, yielding L-zeros to high accuracy. We study in detail the L-function for the modular form based on the Ramanujan τ-function, which is closely related to the bosonic string partition function. The same analysis for a more general class of L-functions is also considered.