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

Abstract

Abstract. We shall prove an unconditional basic result related to the value-distributions of {(L /L)(s, χ)}χ and of {(ζ /ζ)(s + iτ )}τ , where χ runs over Dirichlet characters with prime conductors and τ runs over R. The result asserts that the expected density function common for these distributions are in fact the density function in an appropriate sense. Under the Generalized Riemann hypothesis, stronger results have been proved in our previous articles, but our present result is unconditional.

An M-function associated with Goldbach's problem

“…In 2008, Ihara [16] studied the χ-aspect for L-functions defined on number fields or function fields. His study was then further refined in a series of papers of Ihara and the author [17] [18] [19] [20]. Let us quote a result proved in [18].…”

Nearest neighbor spacing distributions for the zeros of the real or imaginary part of the Riemann xi-function on vertical lines

“…He constructed M σ (z) for σ > 1/2 as the Fourier transform of the Euler product p Mσ,p (z) but test functions in (1.5) are restricted to smooth and compactly supported functions. This restriction for the test functions was relaxed to a wider class of functions by Ihara-Matsumoto [6] in 2011 which was a goal of a series of collaboration works of Ihara and Matsumoto standing on Ihara [3]. In 2008, Ihara [3] studied analytic and arithmetic properties of M σ (z) and Mσ (s) systematically and in detail for σ > 1/2 motivated by a study on Euler-Kronecker constants of global fields.…”

“…This work was refined in Ihara [4]. The formulation of (1.5) in the introduction depends on [3, Theorem 6] and [6].…”

“…We call M σ (z) the M -function according to [3]. The above formula was obtained by Kampen-Wintner [8], Kershner-Wintner [9], Guo [2], Ihara [3] and Ihara-Matsumoto [6] (see Appendix for a construction of M σ (z) and its historical details). If σ > 1, formula (1.5) holds for any continuous function Φ(z) on C.…”

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

“…Ihara [5] discovered that the same type of results can be shown for certain mean values of L ′ /L(s, χ) with respect to characters, where L(s, χ) denotes the Dirichlet (or Hecke) L-function attached to the character χ, including also the function field case. Ihara's work was strengthened, and extended to the log L case, in several joint papers of Ihara and the first author [6], [7], [8], [9]. Recently, Mourtada and Murty [14] obtained an analogous result for the mean value of L ′ /L(s, χ) with respect to discriminants.…”