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Moscow Mathematical Journal

Volume 18, Issue 3, July–September 2018  pp. 437–472.

On M-Functions Associated with Modular Forms

Authors:  Philippe Lebacque (1) and Alexey Zykin (2)
Author institution:(1) Laboratoire de Mathématiques de Besançon, UFR Sciences et techniques 16, route de Gray 25 030 Besançon, France
(2) Laboratoire GAATI, Université de la Polynésie française, BP 6570 – 98702 Faa'a, Tahiti, Polynésie française
National Research University Higher School of Economics
AG Laboratory NRU HSE
Institute for Information Transmission Problems of the Russian Academy of Sciences


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 𝔏(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 L when either χ or f vary. First, for a quasi-character ψ: ℂ → ℂ× we find the limit for the average Avgχψ(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 𝔏(f⊗χ, s) by establishing analytic properties of the above limit function. Third, we study the limit of the harmonic average Avghf 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). 2010 Math. Subj. Class. Primary: 11F11; Secondary: 11M41.

Keywords: L-function, cuspidal newforms, value-distribution, density function.

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