On Certain Mean Values and the Value-Distribution of Logarithms of Dirichlet L-Functions

Abstract

We study the value-distribution of Dirichlet L-functions L(s, χ) in the half-plane σ = s > 1/2. The main result is that a certain average of the logarithm of L(s, χ) with respect to χ, or of the Riemann zeta-function ζ(s) with respect to s, can be expressed as an integral involving a density function, which depends only on σ and can be explicitly constructed. Several mean-value estimates on L-functions are essentially used in the proof in the case 1/2 < σ ≤ 1.

On the Value-Distribution of Logarithmic Derivatives of Dirichlet L-Functions

“…This is a supplement of our former papers [3] and [7]. In [3], we defined and studied the "would-be density function" M σ (w) (σ > 1/2) for the value-distribution of L /L(s, χ) on the complex plane C for certain family of L-functions over any global field (s: fixed with (s) = σ), and established the expected connection under some restrictive hypothesis.…”

“…As for the log L case, such mean value theorems were obtained in [7] to prove Theorem 1.2. It is possible to use the same mean value theorems in our present situation, because L /L can be written as an integral involving log L in the integrand, by using the Cauchy integral formula.…”

“…This was generalized and strengthened in [8] under GRH, the Generalized Riemann hypothesis. In [7] we treated the analogous "would-be" density function M σ (w) for the log L case, and in this case, when the base field is the rational number field Q, we were able to obtain an unconditional result on the expected connection. The purpose of the present paper is to show that a parallel unconditional result for the L /L case over Q can be obtained with but small modifications of the methods used in [7].…”

“…m → ∞), hence the limit average (1.5) remains the same (e.g. the subset "X (f )" in [7], or the subset denoted by X (f ) defined below in Section 2 used for the proof). As regards the additional statement for s = 1, note that X (f, 1) = X(f ).…”

“…In [7], we have constructed a continuous non-negative density function M σ (w) parametrized by σ > 1/2 and established the following theorem.…”