In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zer... more In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1 2. For every prime number p n , we define the sequence X n = q≤p n q q−1 − e γ × log θ(p n), where θ(x) is the Chebyshev function and γ ≈ 0.57721 is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if X n > 0 holds for all primes p n > 2. For every prime number p k > 2, X k > 0 is called the Nicolas inequality. We prove that the Nicolas inequality holds for all primes p n > 2. In this way, we demonstrate that the Riemann hypothesis is true.
We prove a formula relating the Fourier coefficients of a modular form of half-integral weight to... more We prove a formula relating the Fourier coefficients of a modular form of half-integral weight to the special values of L-functions. The form in question is an explicit theta lift from the multiplicative group of an indefinite quaternion algebra over Q. This formula has applications to proving the nonvanishing of this lift and to an explicit version of the Rallis inner product formula.
The aim of this very short note is to give details on Oberdieck derivation. This is an unpublishe... more The aim of this very short note is to give details on Oberdieck derivation. This is an unpublished companion to the work Formal deformations of the algebra of Jacobi forms and Rankin-Cohen brackets by the same authors. We build a natural derivation on Jacobi forms that extends Serre derivation. Our construction has been influenced by a construction of some differential operator by Oberdieck in [Obe14] and hence we shall call this derivation the Oberdieck derivation (see also [DLM00, GK09, MTZ08]). References for the Weierstraß ℘ and ζ functions are [Lan87, Ch. 18], [Sil94, Ch. 1] and [CS17, Ch. 2].
The aims of the conference were: To bring together students and experts in number theory and cryp... more The aims of the conference were: To bring together students and experts in number theory and cryptography. To exchange information and to give reports on the state-of-the art and new results in cryptography and its relation with algebraic number theory. To encourage and stimulate further research concerning the security and implementation of cryptosystems and related areas. To encourage collaboration between mathematicians, computer scientists and engineers in the academic, industry and government sectors. Our workshop was part of a major e ort on the part of Com 2 MaC to promote and support high quality research within the ÿelds of Computational Mathematics. Our deepest gratitude goes to the director, Jin-Ho Kwak, and the steering committee of Com 2 MaC for the generous support of this workshop. The editors sincerely thank sta and students of the center who have helped in the planning of the workshop, and in doing the day-today work which was needed for holding the workshop. Many thanks go to Prof. Sung-Yell Song in Iowa State University and Doctoral student EunJeong Lee who put a lot of e ort into preparing these proceedings. We would also like to acknowledge the Korean Science and Engineering Foundation of POSTECH, and the Mathematics Department of POSTECH, for their ÿnancial support. Without their support, organizing a workshop of this magnitude would have been di cult and, of course, the present collection of articles would not have been realized. Our special thanks go to all of the speakers of the workshop. Their assistance and presence really helped make this workshop a great success. Last, but not least, we extend our thanks to the large number of participants who took part in the workshop. The support and encouragement o ered by them was enormous and contributed a great deal towards the success of the workshop.
... previous :: next. Jacobi forms and the heat operator II. YoungJu Choie. Source: Illinois J. M... more ... previous :: next. Jacobi forms and the heat operator II. YoungJu Choie. Source: Illinois J. Math. Volume 42, Issue 2 (1998), 179-186. Primary Subjects: 11F55. Secondary Subjects: 11F60. Full-text: Access by subscription. PDF File (458 KB). Links and Identifiers. ...
We investigate non-vanishing properties of L(f, s) on the real line, when f is a Hecke eigenform ... more We investigate non-vanishing properties of L(f, s) on the real line, when f is a Hecke eigenform of half-integral weight k + 1 2 on Γ 0 (4).
We derive various identities among the special values of multiple Hecke Lseries. We show that lin... more We derive various identities among the special values of multiple Hecke Lseries. We show that linear combinations of multiple Hecke L-values can be expressed as linear combinations of products of the usual Hecke L-series evaluated at the critical points. The period polynomials introduced here are values of 2-cocycles, whereas the classical period polynomials of elliptic modular forms come from the 1-cocycles. We derive the 2-cycle and the 3-cycle relations among them.
Jacobi-like forms for a discrete subgroup of ΓSL(2-ℝ)are formal power series which generalize Jac... more Jacobi-like forms for a discrete subgroup of ΓSL(2-ℝ)are formal power series which generalize Jacobi forms, and they correspond to certain sequences of modular forms for Γ. Given a modular form Γ, a Jacobilike form can be constructed by using constant multiples of derivatives of as f coefficients, which is known as the Cohen-Kuznetsov lifting of f. We extend Cohen-Kuznetsov liftings to quasimodular forms by determining an explicit formula for a Jacobilike form associated to a quasimodular form
In this paper we show that the invariant polynomial ring of the associated Clifford-Weil group ca... more In this paper we show that the invariant polynomial ring of the associated Clifford-Weil group can be embedded into the ring of Jacobi modular forms over the totally real field, so, therefore, that of Hilbert modular forms over the totally real field. Résumé (Anneau des invariants du groupe de Clifford-Weil, et forme de Jacobi sur un corps to-tallement réel) Dans cet article nous démontrons que l'anneau des polynômes invariants par le groupe de Clifford-Weil peu etre inclus dans l'anneau des formes modulaires de Jacobi sur le corps totalement réel, et donc aussi dans celui des formes modulaires de Hilbert sur le corps totalement réel.
A Jacobi polynomial was introduced by Ozeki. It corresponds to the codes over F2: Later, Bannai a... more A Jacobi polynomial was introduced by Ozeki. It corresponds to the codes over F2: Later, Bannai and Ozeki showed how to construct Jacobi forms with various index using a Jacobi polynomial corresponding to the binary codes. It generalizes Broue-Enguehard map. In this paper, we study Jacobi polynomial which corresponds to the codes over F2f: We show how to construct Jacobi forms with various index over the totally real field. This is one of extension of Broue-Enguehard map.
A certain finiteness result for special values of character twists of Koecher-Maass series attach... more A certain finiteness result for special values of character twists of Koecher-Maass series attached to Siegel cusp of genus g is proved.
ABSTRACT Ten years ago D. Zagier [Proc. Indian Acad. Sci., Math. Sci. 104, 57–75 (1994; Zbl 0806.... more ABSTRACT Ten years ago D. Zagier [Proc. Indian Acad. Sci., Math. Sci. 104, 57–75 (1994; Zbl 0806.11022)] introduced the so-called Rankin-Cohen algebras which consist of differential bilinear operators acting on the graded ring M(Γ) of modular forms with respect to some subgroup Γ of PSL 2 (ℤ). In the paper under review the authors continue their earlier work on Rankin-Cohen brackets with respect to Jacobi forms in the sense of Eichler-Zagier, where the heat operator is involved. In particular they study the algebraic properties of these generalized Rankin-Cohen algebras and give examples generalizing the elliptic case.
Hecke operators on rational period functions on the modular group have beend defined based on mod... more Hecke operators on rational period functions on the modular group have beend defined based on modular integrals. We give a purely algebraic definition of Heeke operators of rational period functions Oil the Heeke groups G(y'2) and G(Y3) and investigate their main properties. This is done with the help of description on the action of Heeke operators on rational period functions on the modular group.
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zer... more In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1 2. For every prime number p n , we define the sequence X n = q≤p n q q−1 − e γ × log θ(p n), where θ(x) is the Chebyshev function and γ ≈ 0.57721 is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if X n > 0 holds for all primes p n > 2. For every prime number p k > 2, X k > 0 is called the Nicolas inequality. We prove that the Nicolas inequality holds for all primes p n > 2. In this way, we demonstrate that the Riemann hypothesis is true.
We prove a formula relating the Fourier coefficients of a modular form of half-integral weight to... more We prove a formula relating the Fourier coefficients of a modular form of half-integral weight to the special values of L-functions. The form in question is an explicit theta lift from the multiplicative group of an indefinite quaternion algebra over Q. This formula has applications to proving the nonvanishing of this lift and to an explicit version of the Rallis inner product formula.
The aim of this very short note is to give details on Oberdieck derivation. This is an unpublishe... more The aim of this very short note is to give details on Oberdieck derivation. This is an unpublished companion to the work Formal deformations of the algebra of Jacobi forms and Rankin-Cohen brackets by the same authors. We build a natural derivation on Jacobi forms that extends Serre derivation. Our construction has been influenced by a construction of some differential operator by Oberdieck in [Obe14] and hence we shall call this derivation the Oberdieck derivation (see also [DLM00, GK09, MTZ08]). References for the Weierstraß ℘ and ζ functions are [Lan87, Ch. 18], [Sil94, Ch. 1] and [CS17, Ch. 2].
The aims of the conference were: To bring together students and experts in number theory and cryp... more The aims of the conference were: To bring together students and experts in number theory and cryptography. To exchange information and to give reports on the state-of-the art and new results in cryptography and its relation with algebraic number theory. To encourage and stimulate further research concerning the security and implementation of cryptosystems and related areas. To encourage collaboration between mathematicians, computer scientists and engineers in the academic, industry and government sectors. Our workshop was part of a major e ort on the part of Com 2 MaC to promote and support high quality research within the ÿelds of Computational Mathematics. Our deepest gratitude goes to the director, Jin-Ho Kwak, and the steering committee of Com 2 MaC for the generous support of this workshop. The editors sincerely thank sta and students of the center who have helped in the planning of the workshop, and in doing the day-today work which was needed for holding the workshop. Many thanks go to Prof. Sung-Yell Song in Iowa State University and Doctoral student EunJeong Lee who put a lot of e ort into preparing these proceedings. We would also like to acknowledge the Korean Science and Engineering Foundation of POSTECH, and the Mathematics Department of POSTECH, for their ÿnancial support. Without their support, organizing a workshop of this magnitude would have been di cult and, of course, the present collection of articles would not have been realized. Our special thanks go to all of the speakers of the workshop. Their assistance and presence really helped make this workshop a great success. Last, but not least, we extend our thanks to the large number of participants who took part in the workshop. The support and encouragement o ered by them was enormous and contributed a great deal towards the success of the workshop.
... previous :: next. Jacobi forms and the heat operator II. YoungJu Choie. Source: Illinois J. M... more ... previous :: next. Jacobi forms and the heat operator II. YoungJu Choie. Source: Illinois J. Math. Volume 42, Issue 2 (1998), 179-186. Primary Subjects: 11F55. Secondary Subjects: 11F60. Full-text: Access by subscription. PDF File (458 KB). Links and Identifiers. ...
We investigate non-vanishing properties of L(f, s) on the real line, when f is a Hecke eigenform ... more We investigate non-vanishing properties of L(f, s) on the real line, when f is a Hecke eigenform of half-integral weight k + 1 2 on Γ 0 (4).
We derive various identities among the special values of multiple Hecke Lseries. We show that lin... more We derive various identities among the special values of multiple Hecke Lseries. We show that linear combinations of multiple Hecke L-values can be expressed as linear combinations of products of the usual Hecke L-series evaluated at the critical points. The period polynomials introduced here are values of 2-cocycles, whereas the classical period polynomials of elliptic modular forms come from the 1-cocycles. We derive the 2-cycle and the 3-cycle relations among them.
Jacobi-like forms for a discrete subgroup of ΓSL(2-ℝ)are formal power series which generalize Jac... more Jacobi-like forms for a discrete subgroup of ΓSL(2-ℝ)are formal power series which generalize Jacobi forms, and they correspond to certain sequences of modular forms for Γ. Given a modular form Γ, a Jacobilike form can be constructed by using constant multiples of derivatives of as f coefficients, which is known as the Cohen-Kuznetsov lifting of f. We extend Cohen-Kuznetsov liftings to quasimodular forms by determining an explicit formula for a Jacobilike form associated to a quasimodular form
In this paper we show that the invariant polynomial ring of the associated Clifford-Weil group ca... more In this paper we show that the invariant polynomial ring of the associated Clifford-Weil group can be embedded into the ring of Jacobi modular forms over the totally real field, so, therefore, that of Hilbert modular forms over the totally real field. Résumé (Anneau des invariants du groupe de Clifford-Weil, et forme de Jacobi sur un corps to-tallement réel) Dans cet article nous démontrons que l'anneau des polynômes invariants par le groupe de Clifford-Weil peu etre inclus dans l'anneau des formes modulaires de Jacobi sur le corps totalement réel, et donc aussi dans celui des formes modulaires de Hilbert sur le corps totalement réel.
A Jacobi polynomial was introduced by Ozeki. It corresponds to the codes over F2: Later, Bannai a... more A Jacobi polynomial was introduced by Ozeki. It corresponds to the codes over F2: Later, Bannai and Ozeki showed how to construct Jacobi forms with various index using a Jacobi polynomial corresponding to the binary codes. It generalizes Broue-Enguehard map. In this paper, we study Jacobi polynomial which corresponds to the codes over F2f: We show how to construct Jacobi forms with various index over the totally real field. This is one of extension of Broue-Enguehard map.
A certain finiteness result for special values of character twists of Koecher-Maass series attach... more A certain finiteness result for special values of character twists of Koecher-Maass series attached to Siegel cusp of genus g is proved.
ABSTRACT Ten years ago D. Zagier [Proc. Indian Acad. Sci., Math. Sci. 104, 57–75 (1994; Zbl 0806.... more ABSTRACT Ten years ago D. Zagier [Proc. Indian Acad. Sci., Math. Sci. 104, 57–75 (1994; Zbl 0806.11022)] introduced the so-called Rankin-Cohen algebras which consist of differential bilinear operators acting on the graded ring M(Γ) of modular forms with respect to some subgroup Γ of PSL 2 (ℤ). In the paper under review the authors continue their earlier work on Rankin-Cohen brackets with respect to Jacobi forms in the sense of Eichler-Zagier, where the heat operator is involved. In particular they study the algebraic properties of these generalized Rankin-Cohen algebras and give examples generalizing the elliptic case.
Hecke operators on rational period functions on the modular group have beend defined based on mod... more Hecke operators on rational period functions on the modular group have beend defined based on modular integrals. We give a purely algebraic definition of Heeke operators of rational period functions Oil the Heeke groups G(y'2) and G(Y3) and investigate their main properties. This is done with the help of description on the action of Heeke operators on rational period functions on the modular group.
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