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.
Proceedings of the American Mathematical Society, Mar 17, 2023
This work addresses the prime number races for non-constant elliptic curves E E over function fi... more This work addresses the prime number races for non-constant elliptic curves E E over function fields. We prove that if r a n k ( E ) > 0 \mathrm {rank}(E) > 0 , then there exist Chebyshev biases towards being negative, and otherwise there exist Chebyshev biases towards being positive. The key input is the convergence of the partial Euler product at the centre, which follows from the Deep Riemann Hypothesis over function fields.
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 .
We refine a formula on values of multiple sine functions at division points. As applications we p... more We refine a formula on values of multiple sine functions at division points. As applications we prove a formula on a sum of reciprocal trigonometric values, and obtain multiple modularity of a three variable modular function, which concerns a generalization of the Dedekind η function.
Proceedings of the Japan Academy. Series A, Mathematical sciences, May 1, 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.
For any congruence subgroup of the modular group, we extend the region of convergence of the Eule... more For any congruence subgroup of the modular group, we extend the region of convergence of the Euler products of the Selberg zeta functions beyond the boundary ℜs = 1, if they are attached with a nontrivial irreducible unitary representation. The region is determined by the size of the lowest eigenvalue of the Laplacian, and it extends to ℜs 3/4 under Selberg's eigenvalue conjecture. More generally, for any unitary representation we establish the relation between the behavior of partial Euler products in the critical strip and the estimate of the error term in the prime geodesic theorem. For the trivial representation, the proof essentially exploits the idea of the celebrated work of Ramanujan. 1991 Mathematics Subject Classification. Primary 11M36; Secondary 11M26 and 11M06. Key words and phrases. Selberg zeta functions and Partial Euler products and Deep Riemann Hypothesis and Explicit formula and Prime geodesic theorem and Selberg's eigenvalue conjecture.
Proceedings of the Japan Academy. Series A, Mathematical sciences, Sep 1, 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.
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 .
Proceedings of the Japan Academy, Series A, Mathematical Sciences
The authors assume the Deep Riemann Hypothesis to prove that a weighted sum of Ramanujan's-functi... more The authors assume the Deep Riemann Hypothesis to prove that a weighted sum of Ramanujan's-function has a bias to being positive. This phenomenon is an analogue of Chebyshev's bias.
Abstract: Two new concepts of zeta functions for schemes over the field of one element are propos... more Abstract: Two new concepts of zeta functions for schemes over the field of one element are proposed. A localization formula and an explicit formula in the affine case are given. This allows for a computation for every scheme.
Reasons for the emergence of Chebyshev's bias were investigated. The Deep Riemann Hypothesis (DRH... more Reasons for the emergence of Chebyshev's bias were investigated. The Deep Riemann Hypothesis (DRH) enables us to reveal that the bias is a natural phenomenon for achieving a well-balanced disposition of the whole sequence of primes, in the sense that the Euler product converges at the center. By means of a weighted counting function of primes, the authors succeed in expressing magnitudes of the deflection by a certain asymptotic formula under the assumption of DRH, which provides a new formulation of Chebyshev's bias. For any Galois extension of global fields and for any element σ in the Galois group, we have established a criterion of the bias of primes whose Frobenius elements are equal to σ under the assumption of DRH. As an application we have obtained a bias toward nonsplitting and non-principle primes in abelian extensions under DRH. In positive characteristic cases, DRH is known, and all these results hold unconditionally.
In this note we present our results on multiple zeta functions with some applications. This is as... more In this note we present our results on multiple zeta functions with some applications. This is asurvey of our papers [KK1, KK2, KK3, KK4]. We also refer to [KK5, KK6] for applications more recently proved.
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2004
We obtain a new bound for the first eigenvalue of the Laplacian for Bianchi manifolds by the meth... more We obtain a new bound for the first eigenvalue of the Laplacian for Bianchi manifolds by the method of Luo, Rudnick and Sarnak. We use a recent result of Kim on symmetric power L-functions. The key idea is to take tensor products of zeta functions, and we report on our recent developments on Kurokawa's multiple zeta functions.
We construct a theory of multiple sine functions generalizing the usual sine function. As applica... more We construct a theory of multiple sine functions generalizing the usual sine function. As applications we have an expression for special values of zeta functions and we calculate gamma factors of Selberg zeta functions.
This article studies the prime number races for non-constant elliptic curves E over function fiel... more This article studies 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 it is shown under the Birch-Swinnerton-Dyer conjecture that there exist Chebyshev biases towards being positive. The proof uses the convergence of the Euler product at the centre implied by the Deep Riemann Hypothesis over function fields.
For a compact Riemann surface M of genus g ≥ 2, we study the functional equations of the Selberg ... more For a compact Riemann surface M of genus g ≥ 2, we study the functional equations of the Selberg zeta functions attached with the Tate motives f. We prove that certain functional equations hold if and only if f has the absolute automorphy.
The second author formulated quantum unique ergodicity for Eisenstein series in the prime level a... more The second author formulated quantum unique ergodicity for Eisenstein series in the prime level aspect in "Equidistribution of Eisenstein series in the level aspect", Comm. Math. Phys. 289, no. 3, 1131-1150 (2009). We point out major errors and propose ideas to correct particular parts of the proofs with partially weakened claims.
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.
Proceedings of the American Mathematical Society, Mar 17, 2023
This work addresses the prime number races for non-constant elliptic curves E E over function fi... more This work addresses the prime number races for non-constant elliptic curves E E over function fields. We prove that if r a n k ( E ) > 0 \mathrm {rank}(E) > 0 , then there exist Chebyshev biases towards being negative, and otherwise there exist Chebyshev biases towards being positive. The key input is the convergence of the partial Euler product at the centre, which follows from the Deep Riemann Hypothesis over function fields.
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 .
We refine a formula on values of multiple sine functions at division points. As applications we p... more We refine a formula on values of multiple sine functions at division points. As applications we prove a formula on a sum of reciprocal trigonometric values, and obtain multiple modularity of a three variable modular function, which concerns a generalization of the Dedekind η function.
Proceedings of the Japan Academy. Series A, Mathematical sciences, May 1, 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.
For any congruence subgroup of the modular group, we extend the region of convergence of the Eule... more For any congruence subgroup of the modular group, we extend the region of convergence of the Euler products of the Selberg zeta functions beyond the boundary ℜs = 1, if they are attached with a nontrivial irreducible unitary representation. The region is determined by the size of the lowest eigenvalue of the Laplacian, and it extends to ℜs 3/4 under Selberg's eigenvalue conjecture. More generally, for any unitary representation we establish the relation between the behavior of partial Euler products in the critical strip and the estimate of the error term in the prime geodesic theorem. For the trivial representation, the proof essentially exploits the idea of the celebrated work of Ramanujan. 1991 Mathematics Subject Classification. Primary 11M36; Secondary 11M26 and 11M06. Key words and phrases. Selberg zeta functions and Partial Euler products and Deep Riemann Hypothesis and Explicit formula and Prime geodesic theorem and Selberg's eigenvalue conjecture.
Proceedings of the Japan Academy. Series A, Mathematical sciences, Sep 1, 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.
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 .
Proceedings of the Japan Academy, Series A, Mathematical Sciences
The authors assume the Deep Riemann Hypothesis to prove that a weighted sum of Ramanujan's-functi... more The authors assume the Deep Riemann Hypothesis to prove that a weighted sum of Ramanujan's-function has a bias to being positive. This phenomenon is an analogue of Chebyshev's bias.
Abstract: Two new concepts of zeta functions for schemes over the field of one element are propos... more Abstract: Two new concepts of zeta functions for schemes over the field of one element are proposed. A localization formula and an explicit formula in the affine case are given. This allows for a computation for every scheme.
Reasons for the emergence of Chebyshev's bias were investigated. The Deep Riemann Hypothesis (DRH... more Reasons for the emergence of Chebyshev's bias were investigated. The Deep Riemann Hypothesis (DRH) enables us to reveal that the bias is a natural phenomenon for achieving a well-balanced disposition of the whole sequence of primes, in the sense that the Euler product converges at the center. By means of a weighted counting function of primes, the authors succeed in expressing magnitudes of the deflection by a certain asymptotic formula under the assumption of DRH, which provides a new formulation of Chebyshev's bias. For any Galois extension of global fields and for any element σ in the Galois group, we have established a criterion of the bias of primes whose Frobenius elements are equal to σ under the assumption of DRH. As an application we have obtained a bias toward nonsplitting and non-principle primes in abelian extensions under DRH. In positive characteristic cases, DRH is known, and all these results hold unconditionally.
In this note we present our results on multiple zeta functions with some applications. This is as... more In this note we present our results on multiple zeta functions with some applications. This is asurvey of our papers [KK1, KK2, KK3, KK4]. We also refer to [KK5, KK6] for applications more recently proved.
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2004
We obtain a new bound for the first eigenvalue of the Laplacian for Bianchi manifolds by the meth... more We obtain a new bound for the first eigenvalue of the Laplacian for Bianchi manifolds by the method of Luo, Rudnick and Sarnak. We use a recent result of Kim on symmetric power L-functions. The key idea is to take tensor products of zeta functions, and we report on our recent developments on Kurokawa's multiple zeta functions.
We construct a theory of multiple sine functions generalizing the usual sine function. As applica... more We construct a theory of multiple sine functions generalizing the usual sine function. As applications we have an expression for special values of zeta functions and we calculate gamma factors of Selberg zeta functions.
This article studies the prime number races for non-constant elliptic curves E over function fiel... more This article studies 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 it is shown under the Birch-Swinnerton-Dyer conjecture that there exist Chebyshev biases towards being positive. The proof uses the convergence of the Euler product at the centre implied by the Deep Riemann Hypothesis over function fields.
For a compact Riemann surface M of genus g ≥ 2, we study the functional equations of the Selberg ... more For a compact Riemann surface M of genus g ≥ 2, we study the functional equations of the Selberg zeta functions attached with the Tate motives f. We prove that certain functional equations hold if and only if f has the absolute automorphy.
The second author formulated quantum unique ergodicity for Eisenstein series in the prime level a... more The second author formulated quantum unique ergodicity for Eisenstein series in the prime level aspect in "Equidistribution of Eisenstein series in the level aspect", Comm. Math. Phys. 289, no. 3, 1131-1150 (2009). We point out major errors and propose ideas to correct particular parts of the proofs with partially weakened claims.
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Papers by Shin-ya Koyama