On the density function for the value-distribution of automorphic L-functions

Abstract

The Bohr-Jessen limit theorem is a probabilistic limit theorem on the value-distribution of the Riemann zeta-function in the critical strip. Moreover their limit measure can be written as an integral involving a certain density function. The existence of the limit measure is now known for a quite general class of zeta-functions, but the integral expression has been proved only for some special cases (such as Dedekind zeta-functions). In this paper we give an alternative proof of the existence of the limit measure for a general setting, and then prove the integral expression, with an explicitly constructed density function, for the case of automorphic L-functions attached to primitive forms with respect to congruence subgroups Γ0(N ). 2010 Mathematics Subject Classification. Primary 11F66, Secondary 11M41.This work is a milestone in the value-distribution theory of ζ(s), and various alternative proofs and related results have been pubished; for example, Jessen and Wintner [8], Borchsenius and Jessen [5], Guo [6], and Ihara and the first author [7].An important problem is to consider the generalization of the Bohr-Jessen theorem. The first author [11] proved that the formula (1.2) can be generalized to a fairly general class of zeta-functions with Euler products. However, (1.3) has not yet been generalized to such a general class. The reason is as follows.The original proof of (1.2) and (1.3) by Bohr and Jessen depends on a geometric theory of certain "infinite sums" of convex curves, developed by themselves [3]. In later articles [8] and [5], the effect of the convexity of curves was embodied in a certain inequality due to Jessen and Wintner [8, Theorem 13]. Using this method, the Bohr-Jessen theory was generalized to Dirichlet L-functions (Joyner [9]) and Dedekind zeta-functions of Galois number fields (the first author [12]). These generalizations are possible because these zeta-functions have "convex" Euler products in the sense of [11, Section 5]. But this convexity cannot be expected for more general zetafunctions.In [11], the first author developed a method of proving (1.2) without using any convexity, so succeeded in generalizing the theory. However, the method in [11] cannot give a generalization of (1.3).So far, there is no proof of (1.3) or its analogues without using the convexity, or the Jessen-Wintner type of inequalities. For example, [7] gives a different argument of constructing the density functions for Dirichlet L-functions, but the argument in [7] also depends on the Jessen-Wintner inequality.In [14] [15], the first author obtained certain quantitative results on the value-distribution of Dedekind zeta-functions of non-Galois fields and Hecke L-functions of ideal class characters, whose Euler products are not convex. But in these cases, they are "not so far" from the case of Dedekind zetafunctions of Galois fields. In fact, a simple generalization of the Jessen-Wintner inequality is proved ( [15, Lemma 2]) and is essentially used in the proof.Actually, analyzing the proof of [8, Theorems 12, 13...

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

“…Then P f (ε) is of positive density (M. R. Murty [34] for the full modular case, and M. R. Murty and V. K. Murty [35] for any level N ). In [28], we observed geometric behavior of the corresponding curves and proved Lemma 5.4 ([28]). If p n ∈ P f (ε) and n is sufficiently large, then…”

“…The basic structure of the proof of Theorem 5.1 in [28], which we briefly outline here, is along the line similar to [24]. Actually in [28], we are working in more general situation, that is in the class M . Let ϕ ∈ M .…”

“…The method in [24] is to use Lévy's convergence theorem, again in probability theory. This method can also be applied to general ϕ ∈ M , which is pointed out in [25] and a sketch of the argument in the general case is described in [28].…”

“…Automorphic L-functions are typical examples for which the corresponding curves are not always convex, so it is important how to generalize the part (ii) of Theorem 1.1 to the case of automorphic L-functions L(f, s). This has been done in [28].…”

“…where z n (θ n ; ϕ) is defined by (2.6) and z, w = ℜzℜw + ℑzℑw. In [28], we prove the following Lemma 5.3 ([28]). If there are at least five n's for which…”