2010
DOI: 10.1093/qmath/haq002
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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.

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Cited by 24 publications
(76 citation statements)
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“…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.…”
Section: Introduction and Statement Of The Resultsmentioning
confidence: 81%
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“…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.…”
Section: Introduction and Statement Of The Resultsmentioning
confidence: 81%
“…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.…”
Section: Introduction and Statement Of The Resultsmentioning
confidence: 96%
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