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Yasufumi Hashimoto

Publications and source records attributed to Yasufumi Hashimoto.

14 recordsLinked to original sources
arXiv2023

Universality theorems of the Selberg zeta functions for arithmetic groups

After Voronin proved the universality theorem of the Riemann zeta function in the 1970s, universality theorems have been proposed for various zeta and L-functions. Drungilas-Garunkstis-Kacenas' work at 2013 on the universality theorem of the Selberg zeta function for the modular group is one of them and is probably the first universality theorem of the zeta function of order greater than one. Recently, Mishou (2021) extended it by proving the joint universality theorem for the principal congruence subgroups. In the present paper, we further extend these works by proving the (joint) universality theorem for subgroups of the modular group and co-compact arithmetic groups derived from indefinite quaternion algebras, which is available in the region wider than the regions in the previous two works.

math.NT↗
arXiv2021

Square integrals of the logarithmic derivatives of Selberg's zeta functions in the critical strip

In our previous work (https://doi.org/10.1002/mana.202000268, Math. Nachr., 2021), we proposed an upper bound of the logarithmic derivative of Selberg's zeta function for the modular groups in the critical strip. The present paper studies the growth of its square integral for the modular group, co-compact arithmetic groups derived from indefinite quaternion algebras and their subgroups.

math.NT↗
arXiv2019

Prime geodesic theorem for the modular group

Soundararajan and Young (2013) proposed a new approach to improve the error term of the prime geodesic theorem for the modular group and actually obtained the exponent 25/36 of the error term. In the present paper, we sharpen it to 9/13.

math.NT↗
arXiv2015

Distributions of length multiplicities for negatively curved locally symmetric Riemannian manifolds

The aim of the present paper is to study the distributions of the length multiplicities for negatively curved locally symmetric Riemannian manifolds. In Theorem 2.1, we give upper bounds of the length multiplicities and the square sums of them for general (not necessarily compact) cases. Furthermore in Theorem 2.2, we obtain more precise estimates of the length multiplicities and the power sums of them for arithmetic surfaces whose fundamental groups are congruence subgroups of the modular group.

math.SP↗
arXiv2012

Correlations of multiplicities in length spectra for congruence subgroups

Bogomolny-Leyvraz-Schmit (1996) and Peter (2002) proposed an asymptotic formula for the correlation of the multiplicities in length spectrum on the modular surface, and Lukianov (2007) extended its asymptotic formula to the Riemann surfaces derived from the congruence subgroup $\Gam_0(n)$ and the quaternion type co-compact arithmetic groups. The coefficients of the leading terms in these asymptotic formulas are described in terms of Euler products over prime numbers, and they appear in eigenvalue statistic formulas found by Rudnick (2005) and Lukianov (2007) for the Laplace-Beltrami operators on the corresponding Riemann surfaces. In the present paper, we further extend their asymptotic formulas to the higher level correlations of the multiplicities for any congruence subgroup of the modular group.

math.NT↗
arXiv2010

Asymptotic formulas for class number sums of indefinite binary quadratic forms in arithmetic progressions

It is known that there is a one-to-one correspondence between equivalence classes of primitive indefinite binary quadratic forms and primitive hyperbolic conjugacy classes of the modular group. Due to such a correspondence, Sarnak obtained the asymptotic formula for the class number sum in order of the fundamental unit by using the prime geodesic theorem for the modular group. In the present paper, we propose asymptotic formulas of the class number sums over discriminants in arithmetic progressions. Since there are relations between the arithmetic properties of the discriminants and the conjugacy classes in the finite groups given by the modular group and its congruence subgroups, we can get the desired asymptotic formulas by arranging the Tchebotarev-type prime geodesic theorem. While such asymptotic formulas were already given by Raulf, the approaches are quite different, the expressions of the leading terms of our asymptotic formulas are simpler and the estimates of the reminder terms are sharper.

math.NT↗
arXiv2008

Prime and zero distributions for meromorphic Euler products

The aim of the present paper is to study the relations between the prime distribution and the zero distribution for generalized zeta functions which are expressed by Euler products and is analytically continued as meromorphic functions of finite order. In this paper, we give an inequality between the order of the zeta function as a meromorphic function and the growth of the multiplicity in the prime distribution.

math.NT↗
arXiv2007

Splitting density for lifting about discrete groups

We study splitting densities of primitive elements of a discrete subgroup of a connected non-compact semisimple Lie group of real rank one with finite center in another larger such discrete subgroup. When the corresponding cover of such a locally symmetric negatively curved Riemannian manifold is regular, the densities can be easily obtained from the results due to Sarnak or Sunada. Our main interest is a case where the covering is not necessarily regular. Specifically, for the cases of the modular group and its congruence subgroups, we determine the splitting densities explicitly. As an application, we study analytic properties of the zeta function defined by the Euler product over elements consisting all primitive elements which satisfy a certain splitting law for a given lifting.

math.NT↗
arXiv2006

Partial zeta functions

In the present manuscript, we study analytic properties of zeta functions defined by partial Euler products.

math.NT↗
arXiv2006

Arithmetic expressions of Selberg's zeta functions for congruence subgroups

In Sarnak's paper, it was proved that the Selberg zeta function for SL(2,Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL(2,Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem, the asymptotic behavior of the sum of the class number.

math.RT↗
arXiv2005

Hierarchy of the Selberg zeta functions

We introduce a Selberg type zeta function of two variables which interpolates several higher Selberg zeta functions. The analytic continuation, the functional equation and the determinant expression of this function via the Laplacian on a Riemann surface are obtained.

math-ph↗