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

Matsumoto, Kohji; Umegaki, Yumiko

doi:10.48550/arxiv.1603.07436

Cited by 2 publications

(3 citation statements)

References 6 publications

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“…Their approach is based on [10] rather than on [11], though the employed techniques are remarkably close to the ones we apply in §5. The results of Matsumoto and Umegaki are complementary to ours, since the case of Sym 1 f = f, which is the main subject of our paper, could not be treated in [26].…”

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confidence: 55%

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On M-Functions Associated with Modular Forms

Lebacque¹,

Zykin²

2018

MMJ

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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).

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confidence: 55%

“…Finally, let us quote a still more recent preprint by K. Matsumoto and Y. Umegaki [26] that treats similar questions for differences of logarithms of two symmetric power L-functions under the assumption of the GRH. Their approach is based on [10] rather than on [11], though the employed techniques are remarkably close to the ones we apply in §5.…”

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confidence: 99%

On M-Functions Associated with Modular Forms

Lebacque¹,

Zykin²

2018

MMJ

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“…Moreover, the condition for σ was weakened to σ > 1/2 unconditionally under some revisions of the meanings of averages and restrictions for a class of test functions [9,10,12]. Today there are several variations of the results of Ihara-Matsumoto studied by Ihara-Matsumoto [11], Lebacque-Zykin [16], Matsumoto-Umegaki [21], and Mourtada-Murty [23], however, a generalization of the case (C) to an arbitrary number field is not known by a technical difficulty as well as the formula (1.2). We solve such difficulties by the method of Guo [4] below.…”

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confidence: 99%

On the value-distributions of logarithmic derivatives of Dedekind zeta functions

Mine

2017

Preprint

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We study the distributions of values of the logarithmic derivatives of the Dedekind zeta functions on a fixed vertical line. The main object is determining and investigating the density functions of such value-distributions for any algebraic number field. We construct the density functions as the Fourier inverse transformations of certain functions represented by infinite products that come from the Euler products of the Dedekind zeta functions.

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

Cited by 2 publications

References 6 publications

On M-Functions Associated with Modular Forms

On M-Functions Associated with Modular Forms

On the value-distributions of logarithmic derivatives of Dedekind zeta functions

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