On M-Functions Associated with Modular Forms

Abstract

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 Avg χ ψ(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 Avg h 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).

An M-function associated with Goldbach's problem

“…Since then, various analogues of Theorem 2.1 were discovered by Mourtada and Murty [35], Akbary and Hamieh [1], Lebacque and Zykin [24], Matsumoto and Umegaki [29], Mine [32] [33] [34], and so on.…”

On the value-distribution of the difference between logarithms of two symmetric power L-functions

“…More difficult is the case of the level aspect. So far there are two attempts in this direction, the aforementioned paper of Lebacque and Zykin [22], and an article of the authors [27]. Here we briefly mention the results proved in [27].…”

“…(The theorem is shown only for µ ≥ 3.) Lebacque and Zykin [22] studied log L(f, s) and (L ′ /L)(f, s) along the line of [13], and obtained a result analogous to [13,Theorem 1]. However their argument also does not arrive at the limit theorem for log L(f, s) or (L ′ /L)(f, s) of the form (3.1) or (3.4).…”

“…The twisted L-function L(f ⊗χ, s) is defined by replacing p −s by χ(p)p −s on each local factor. Lebacque and Zykin [22] developed the theory similar to [13] for L(f ⊗ χ, s), and proved the limit formulas corresponding to (3.1) and (3.4).…”

Discrete value-distribution of Artin $L$-functions associated with cubic fields

“…These studies were based on the result of Conrey-Duke-Farmer [6] or Serre [24] on the distribution of the eigenvalues of Hecke operators. On the other hand, Lebacque-Zykin [19] and Matsumoto-Umegaki [21] obtained analogues of Theorem 1.1 for L(s, f ) by adapting the Petersson formula.…”