We prove the quantum unique ergodicity conjecture for Eisenstein series over function fields in t... more We prove the quantum unique ergodicity conjecture for Eisenstein series over function fields in the level aspect. Adapting the machinery of Luo-Sarnak (1995), we employ the spectral decomposition and handle the cuspidal and Eisenstein contributions separately.
Transactions of the American Mathematical Society, Mar 1, 1991
We show that for PSL(2, R) and its congruence subgroup, the Selberg zeta function with its gamma ... more We show that for PSL(2, R) and its congruence subgroup, the Selberg zeta function with its gamma factors is expressed as the determinant of the Laplacian. All the gamma factors are calculated explicitly. We also give an explicit computation to the contribution of the continuous spectrum to the determinant of the Laplacian.
Transactions of the American Mathematical Society, 1992
This paper is the PSL(2, C)-version of Part I. We show that for PSL(2, C) and its subgroup PSL(2,... more This paper is the PSL(2, C)-version of Part I. We show that for PSL(2, C) and its subgroup PSL(2, O), the Selberg zeta function with its gamma factors is expressed as the determinant of the Laplacians, where O is the integer ring of an imaginary quadratic field. All the gamma factors are calculated explicitly. We also give an explicit computation to the contribution of the continuous spectrum to the determinant of the Laplacian.
ring of an imaginary quadratic field K of class number one. 1. Introduction. We first explain the... more ring of an imaginary quadratic field K of class number one. 1. Introduction. We first explain the whole picture around the two fields, number theory and quantum chaos. Number theorists study the theory of zeta functions, where one of their chief concerns is to estimate the size of zeta functions such as $| \zeta(\frac{1}{2}+it)|$ along the critical line. Atrivial estimate can be obtained from the convexity principle in the general theory of complex functions. We call it the convexity bound. Any estimate breaking the convexity bound is caUed asubconvexity bound. To obtain any subconvexity estimate is significant in number theory. On the other hand in quantum mechanics or spectral geometry, there is afield called quantum chaos, where they study various problems
The Riemann Hypothesis, one of the Millennium Problems, was formulated by Bernhard Riemann in 185... more The Riemann Hypothesis, one of the Millennium Problems, was formulated by Bernhard Riemann in 1859 as part of his attempts to understand how primes are distributed along the number line. In this expository article, we consider the Deep Riemann Hypothesis for the general linear group GL n , which concerns the convergence of partial Euler products of L-functions on the critical line. Applications of the Deep Riemann Hypothesis include an improvement of the quality of the error term in the prime number theorem equivalent to the Riemann Hypothesis and the verification of Chebyshev's bias for Satake parameters for GL n .
This work considers the prime number races for non-constant elliptic curves E over function field... more This work considers the prime number races for non-constant elliptic curves E over function fields. We prove that if rank(E) > 0, then there exist Chebyshev biases towards being negative, and otherwise there exist Chebyshev biases towards being positive. The main innovation entails the convergence of the partial Euler product at the centre that follows from the Deep Riemann Hypothesis over function fields.
We introduce the notion of absolute modular forms and give examples expressed in terms of higher ... more We introduce the notion of absolute modular forms and give examples expressed in terms of higher derivatives of the multiple sine functions. We also prove a certain identity between them, which suggests a graded structure of the space of absolute modular forms of weight 3.
A new interpretation of zeta functions is given for F1-schemes which do not satisfy Soul\'e&#... more A new interpretation of zeta functions is given for F1-schemes which do not satisfy Soul\'e's condition. Functional equations for reductive groups are computed and a new definition of zeta functions attached to more general counting functions is given which is based on regularization and puts on an equal footing F1-theory on the one hand and spectral theory of Laplace operators on manifolds on the other.
We estimate the number of relatively r-prime lattice points in K m {K^{m}} with their components ... more We estimate the number of relatively r-prime lattice points in K m {K^{m}} with their components having a norm less than x, where K is a number field. The error terms are estimated in terms of x and the discriminant D of the field K, as both x and D grow. The proof uses the bounds of Dedekind zeta functions. We obtain uniform upper bounds as K runs through number fields of any degree under assuming the Lindelöf hypothesis. We also show unconditional results for abelian extensions with a degree less than or equal to 6.
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 1991
1. Introduction. Nowadays Selberg zeta functions are defined for semi-simple Lie groups G of real... more 1. Introduction. Nowadays Selberg zeta functions are defined for semi-simple Lie groups G of real rank one and its discrete subgroups F, since Selberg defined it first in [11] in 1956. Two ways are known for
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2002
We obtain an arithmetic expression of the Selberg zeta function for cocompact Fuchsian group defi... more We obtain an arithmetic expression of the Selberg zeta function for cocompact Fuchsian group defined via an indefinite division quaternion algebra over Q. As application to the prime geodesic theorem, we prove certain uniformity of the distribution.
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2002
We consider the Hecke L-function L(s, λ m) of the imaginary quadratic field Q(i) with the m-th Gr... more We consider the Hecke L-function L(s, λ m) of the imaginary quadratic field Q(i) with the m-th Grossencharacter λ m. We obtain the universality property of L(s, λ m) as both m and t = Im(s) go to infinity.
Casimir Force, Casimir Operators and the Riemann Hypothesis, 2010
We prove that the absolute zeta functions for finite abelian monoids have positive Casimir energy... more We prove that the absolute zeta functions for finite abelian monoids have positive Casimir energy if and only if its order is divisible by 4.
We prove the quantum unique ergodicity conjecture for Eisenstein series over function fields in t... more We prove the quantum unique ergodicity conjecture for Eisenstein series over function fields in the level aspect. Adapting the machinery of Luo-Sarnak (1995), we employ the spectral decomposition and handle the cuspidal and Eisenstein contributions separately.
Transactions of the American Mathematical Society, Mar 1, 1991
We show that for PSL(2, R) and its congruence subgroup, the Selberg zeta function with its gamma ... more We show that for PSL(2, R) and its congruence subgroup, the Selberg zeta function with its gamma factors is expressed as the determinant of the Laplacian. All the gamma factors are calculated explicitly. We also give an explicit computation to the contribution of the continuous spectrum to the determinant of the Laplacian.
Transactions of the American Mathematical Society, 1992
This paper is the PSL(2, C)-version of Part I. We show that for PSL(2, C) and its subgroup PSL(2,... more This paper is the PSL(2, C)-version of Part I. We show that for PSL(2, C) and its subgroup PSL(2, O), the Selberg zeta function with its gamma factors is expressed as the determinant of the Laplacians, where O is the integer ring of an imaginary quadratic field. All the gamma factors are calculated explicitly. We also give an explicit computation to the contribution of the continuous spectrum to the determinant of the Laplacian.
ring of an imaginary quadratic field K of class number one. 1. Introduction. We first explain the... more ring of an imaginary quadratic field K of class number one. 1. Introduction. We first explain the whole picture around the two fields, number theory and quantum chaos. Number theorists study the theory of zeta functions, where one of their chief concerns is to estimate the size of zeta functions such as $| \zeta(\frac{1}{2}+it)|$ along the critical line. Atrivial estimate can be obtained from the convexity principle in the general theory of complex functions. We call it the convexity bound. Any estimate breaking the convexity bound is caUed asubconvexity bound. To obtain any subconvexity estimate is significant in number theory. On the other hand in quantum mechanics or spectral geometry, there is afield called quantum chaos, where they study various problems
The Riemann Hypothesis, one of the Millennium Problems, was formulated by Bernhard Riemann in 185... more The Riemann Hypothesis, one of the Millennium Problems, was formulated by Bernhard Riemann in 1859 as part of his attempts to understand how primes are distributed along the number line. In this expository article, we consider the Deep Riemann Hypothesis for the general linear group GL n , which concerns the convergence of partial Euler products of L-functions on the critical line. Applications of the Deep Riemann Hypothesis include an improvement of the quality of the error term in the prime number theorem equivalent to the Riemann Hypothesis and the verification of Chebyshev's bias for Satake parameters for GL n .
This work considers the prime number races for non-constant elliptic curves E over function field... more This work considers the prime number races for non-constant elliptic curves E over function fields. We prove that if rank(E) > 0, then there exist Chebyshev biases towards being negative, and otherwise there exist Chebyshev biases towards being positive. The main innovation entails the convergence of the partial Euler product at the centre that follows from the Deep Riemann Hypothesis over function fields.
We introduce the notion of absolute modular forms and give examples expressed in terms of higher ... more We introduce the notion of absolute modular forms and give examples expressed in terms of higher derivatives of the multiple sine functions. We also prove a certain identity between them, which suggests a graded structure of the space of absolute modular forms of weight 3.
A new interpretation of zeta functions is given for F1-schemes which do not satisfy Soul\'e&#... more A new interpretation of zeta functions is given for F1-schemes which do not satisfy Soul\'e's condition. Functional equations for reductive groups are computed and a new definition of zeta functions attached to more general counting functions is given which is based on regularization and puts on an equal footing F1-theory on the one hand and spectral theory of Laplace operators on manifolds on the other.
We estimate the number of relatively r-prime lattice points in K m {K^{m}} with their components ... more We estimate the number of relatively r-prime lattice points in K m {K^{m}} with their components having a norm less than x, where K is a number field. The error terms are estimated in terms of x and the discriminant D of the field K, as both x and D grow. The proof uses the bounds of Dedekind zeta functions. We obtain uniform upper bounds as K runs through number fields of any degree under assuming the Lindelöf hypothesis. We also show unconditional results for abelian extensions with a degree less than or equal to 6.
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 1991
1. Introduction. Nowadays Selberg zeta functions are defined for semi-simple Lie groups G of real... more 1. Introduction. Nowadays Selberg zeta functions are defined for semi-simple Lie groups G of real rank one and its discrete subgroups F, since Selberg defined it first in [11] in 1956. Two ways are known for
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2002
We obtain an arithmetic expression of the Selberg zeta function for cocompact Fuchsian group defi... more We obtain an arithmetic expression of the Selberg zeta function for cocompact Fuchsian group defined via an indefinite division quaternion algebra over Q. As application to the prime geodesic theorem, we prove certain uniformity of the distribution.
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2002
We consider the Hecke L-function L(s, λ m) of the imaginary quadratic field Q(i) with the m-th Gr... more We consider the Hecke L-function L(s, λ m) of the imaginary quadratic field Q(i) with the m-th Grossencharacter λ m. We obtain the universality property of L(s, λ m) as both m and t = Im(s) go to infinity.
Casimir Force, Casimir Operators and the Riemann Hypothesis, 2010
We prove that the absolute zeta functions for finite abelian monoids have positive Casimir energy... more We prove that the absolute zeta functions for finite abelian monoids have positive Casimir energy if and only if its order is divisible by 4.
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Papers by Shin-ya Koyama