2017/02/24 by Philippe Lebacque, Lebacque, Philippe, Alexey Zykin +1
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1702.07610
arxiv created 2017/02/24 · openalex publication_date 2017/02/24 · arxiv updated 2017/02/27 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
Let f be a primitive cusp form of weight k and level N, let χ be a Dirichlet character of conductor coprime with N, and let \mathfrakL(f⊗ χ, s) denote either log L(f⊗ χ, s) or (L'/L)(f⊗ χ, s). In this article we study the distribution of the values of \mathfrakL when either χ or f vary. First, for a quasi-character ψ\colon ℂ → ℂ^× we find the limit for the average Avg_χψ(L(f⊗χ, s)), when f is fixed and χ varies through the set of characters with prime conductor that tends to infinity. Second, we prove an equidistribution result for the values of \mathfrakL(f⊗ χ,s) by establishing analytic properties of the above limit function. Third, we study the limit of the harmonic average Avgh_f ψ(L(f, s)), when f runs through the set of primitive cusp forms of given weight k and level N→ ∞. Most of the results are obtained conditionally on the Generalized Riemann Hypothesis for L(f⊗χ, s).