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Masahiro Mine

Publications and source records attributed to Masahiro Mine.

12 recordsLinked to original sources
arXiv2026

Connection between the Riemann zeta-function and random matrices via hyperfunctions

Bohr pioneered the study of the statistical behavior of the Riemann zeta-function. A classical result by Bohr and Jessen revealed that the values of the Riemann zeta-function to the right of the critical line behave like a random variable. We now propose to extend Bohr's theory to the stage of hyperfunctions. In this paper, we introduce two random hyperfunctions: one is associated with the values of the Riemann zeta-function on the critical line, and the other is associated with the characteristic polynomial of a random matrix from the circular unitary ensemble. We then derive a relationship between these random hyperfunctions which is consistent with the Keating-Snaith conjecture on the moments of the Riemann zeta-function.

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arXiv2025

An upper bound for the number of smooth values of a polynomial and its applications

We prove a new upper bound for the number of smooth values of a polynomial with integer coefficients. This improves Timofeev's previous result unless the polynomial is a product of linear polynomials with integer coefficients. As an application, we provide another proof for a result of Cassels which was used to prove that the Hurwitz zeta-function with algebraic irrational parameter has infinitely many zeros on the domain of convergence. We also apply the main result to a problem on primitive divisors of quadratic polynomials.

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arXiv2024

New developments toward the Gonek Conjecture on the Hurwitz zeta-function

In this paper, we prove a version of the universality theorem for the Hurwitz zeta-function in the case where the parameter is algebraic and irrational. Then we apply the result to show that many of such Hurwitz zeta-functions have infinitely many zeros in the right half of the critical strip.

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arXiv2023

On the value-distribution of the logarithms of symmetric power L-functions in the level aspect

We consider the value distribution of logarithms of symmetric power L-functions associated with newforms of even weight and prime power level. In the symmetric square case, under certain plausible analytical conditions, we prove that certain averages of those values in the level aspect, involving continuous bounded or Riemann integrable test functions, can be written as integrals involving a density function (the "M-function") which is related with the Sato-Tate measure. Moreover, even in the case of general symmetric power L-functions, we show the same type of formula when for some special type of test functions. We see that a kind of parity phenomenon of the density function exists.

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arXiv2022

A random variable related to the Hurwitz zeta-function with algebraic parameter

In this paper, we introduce a certain random variable closely related to the value-distribution of the Hurwitz zeta-function with algebraic parameter. We prove a version of the limit theorem, where the limit measure is presented by the law of this random variable. Then we apply it to show that any complex number can be approximated by values of the Hurwitz zeta-function for algebraic irrational parameters but with finite exceptions.

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arXiv2022

The value-distribution of Artin $L$-functions associated with cubic fields in conductor aspect

Arising from the factorizations of Dedekind zeta-functions of cubic fields, we obtain Artin $L$-functions of certain two-dimensional representations. In this paper, we study the value-distribution of such Artin $L$-functions for families of non-Galois cubic fields in conductor aspect. We prove that various mean values of the Artin $L$-functions are represented by integrals involving a density function which can be explicitly constructed. By the class number formula, the result is applied to the study on the distribution of class numbers of cubic fields.

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arXiv2021

The density function for the value-distribution of the Lerch zeta-function and its applications

The probabilistic study of the value-distributions of zeta-functions is one of the modern topics in analytic number theory. In this paper, we study a certain probability measure related to the value-distribution of the Lerch zeta-function. We prove that it has a density function which we can explicitly construct. Moreover, we prove an asymptotic formula for the number of zeros of the Lerch zeta-function on the right side of the critical line, whose main term is associated with the density function.

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arXiv2021

On certain mean values of logarithmic derivatives of $L$-functions and the related density functions

We study some "density function" related to the value-distribution of $L$-functions. The first example of such a density function was given by Bohr and Jessen in 1930s for the Riemann zeta-function. In this paper, we construct the density function in a wide class of $L$-functions. We prove that certain mean values of $L$-functions in the class are represented as integrals involving the related density functions.

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arXiv2021

Probability density functions attached to random Euler products for automorphic $L$-functions

In this paper, we study the value-distributions of $L$-functions of holomorphic primitive cusp forms in the level aspect. We associate such automorphic $L$-functions with probabilistic models called the random Euler products. First, we prove the existence of probability density functions attached to the random Euler products. Then various mean values of automorphic $L$-functions are expressed as integrals involving the density functions. Moreover, we estimate the discrepancies between the distributions of values of automorphic $L$-functions and those of the random Euler products.

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arXiv2017

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

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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