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Negative first moment of quadratic twists of L-functions
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-18 DOI: 10.1016/j.jnt.2025.01.003
Peng Gao , Liangyi Zhao
{"title":"Negative first moment of quadratic twists of L-functions","authors":"Peng Gao ,&nbsp;Liangyi Zhao","doi":"10.1016/j.jnt.2025.01.003","DOIUrl":"10.1016/j.jnt.2025.01.003","url":null,"abstract":"<div><div>We evaluate asymptotically the negative first moment at points larger than 1/2 of the family of quadratic twists of automorphic <em>L</em>-functions using multiple Dirichlet series under the generalized Riemann hypothesis and the Ramanujan-Petersson conjecture.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 389-406"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Approximation of L-functions associated to Hecke cusp eigenforms
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-18 DOI: 10.1016/j.jnt.2025.01.014
An Huang, Kamryn Spinelli
{"title":"Approximation of L-functions associated to Hecke cusp eigenforms","authors":"An Huang,&nbsp;Kamryn Spinelli","doi":"10.1016/j.jnt.2025.01.014","DOIUrl":"10.1016/j.jnt.2025.01.014","url":null,"abstract":"<div><div>We derive a family of approximations for L-functions of Hecke cusp eigenforms, according to a recipe first described by Matiyasevich for the Riemann xi function. We show that these approximations converge to the true L-function and point out the role of an equidistributional notion in ensuring the approximation is well-defined, and along the way we demonstrate error formulas which may be used to investigate analytic properties of the L-function and its derivatives, such as the locations and orders of zeros. Together with the Euler product expansion of the L-function, the family of approximations also encodes some of the key features of the L-function such as its functional equation. As an example, we apply this method to the L-function of the modular discriminant and demonstrate that the approximation successfully locates zeros of the L-function on the critical line. Finally, we derive via Mellin transforms a convolution-type formula which leads to precise error bounds in terms of the incomplete gamma function. This formula can be interpreted as an alternative definition for the approximation and sheds light on Matiyasevich's procedure.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"272 ","pages":"Pages 60-84"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On some unramified families of motivic Euler sums 论动机欧拉和的一些非ramified族
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-18 DOI: 10.1016/j.jnt.2024.11.009
Ce Xu , Jianqiang Zhao
{"title":"On some unramified families of motivic Euler sums","authors":"Ce Xu ,&nbsp;Jianqiang Zhao","doi":"10.1016/j.jnt.2024.11.009","DOIUrl":"10.1016/j.jnt.2024.11.009","url":null,"abstract":"<div><div>It is well known that sometimes Euler sums (i.e., alternating multiple zeta values) can be expressed as <span><math><mi>Q</mi></math></span>-linear combinations of multiple zeta values (MZVs). In her thesis Glanois presented a criterion for motivic Euler sums to be unramified, namely, expressible as <span><math><mi>Q</mi></math></span>-linear combinations of motivic MZVs. By applying this criterion we present a few families of such unramified motivic Euler sums in two groups. In one such group we can further prove the explicit identities relating the motivic Euler sums to the motivic MZVs, under the assumption that the analytic version of such identities holds. We also propose a conjecture concerning a vast family of unramified motivic Euler sums that simultaneously generalizes all the results contained in this paper.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"272 ","pages":"Pages 85-112"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Kummer theory for function fields
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2025.01.004
Félix Baril Boudreau, Antonella Perucca
{"title":"Kummer theory for function fields","authors":"Félix Baril Boudreau,&nbsp;Antonella Perucca","doi":"10.1016/j.jnt.2025.01.004","DOIUrl":"10.1016/j.jnt.2025.01.004","url":null,"abstract":"<div><div>We develop Kummer theory for algebraic function fields in finitely many transcendental variables. We consider any finitely generated Kummer extension (possibly, over a cyclotomic extension) of an algebraic function field, and describe the structure of its Galois group. Our results show in a precise sense how the questions of computing the degrees of these extensions and of computing the group structures of their Galois groups reduce to the corresponding questions for the Kummer extensions of their constant fields.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 504-525"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a twisted Jacquet module of GL(2n) over a finite field
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2024.12.006
Kumar Balasubramanian , Abhishek Dangodara , Himanshi Khurana
{"title":"On a twisted Jacquet module of GL(2n) over a finite field","authors":"Kumar Balasubramanian ,&nbsp;Abhishek Dangodara ,&nbsp;Himanshi Khurana","doi":"10.1016/j.jnt.2024.12.006","DOIUrl":"10.1016/j.jnt.2024.12.006","url":null,"abstract":"<div><div>Let <em>F</em> be a finite field and <span><math><mi>G</mi><mo>=</mo><mi>GL</mi><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>. Let <em>π</em> be an irreducible cuspidal representation of <em>G</em>. In this paper, we give an explicit description of the structure of the twisted Jacquet module of <em>π</em> corresponding to a degenerate character of <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> of rank one.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 458-474"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Continuous lower bounds for moments of the mixed product of twisted L-functions
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2024.12.003
Guohua Chen , Weiping Li , Tianze Wang
{"title":"Continuous lower bounds for moments of the mixed product of twisted L-functions","authors":"Guohua Chen ,&nbsp;Weiping Li ,&nbsp;Tianze Wang","doi":"10.1016/j.jnt.2024.12.003","DOIUrl":"10.1016/j.jnt.2024.12.003","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> be two distinct cuspidal holomorphic Hecke eigenforms of even weight <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>κ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> for the modular group <span><math><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span> respectively. Let <span><math><mi>L</mi><mo>(</mo><mi>s</mi><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊗</mo><mi>χ</mi><mo>)</mo><mo>,</mo><mi>L</mi><mo>(</mo><mi>s</mi><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊗</mo><mi>χ</mi><mo>)</mo></math></span> denote twisted <em>L</em>-functions associated to <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> twisted by a primitive Dirichlet character <em>χ</em> modulo <em>q</em> respectively. In this paper, we obtain sharp lower bounds for all positive real <em>k</em>-th <span><math><mo>(</mo><mi>k</mi><mo>&gt;</mo><mn>1</mn><mo>)</mo></math></span> moments of the mixed product of these two twisted <em>L</em>-functions at the central value, which extend the previous results.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 438-457"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the distribution of Ω(n)−ω(n)
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2024.12.002
Biao Wang
{"title":"On the distribution of Ω(n)−ω(n)","authors":"Biao Wang","doi":"10.1016/j.jnt.2024.12.002","DOIUrl":"10.1016/j.jnt.2024.12.002","url":null,"abstract":"<div><div>Let <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the number of all prime factors and distinct prime factors of <em>n</em>, respectively. In 1955, Rényi found the density for the numbers <em>n</em> such that <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>−</mo><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>k</mi></math></span> for all integers <span><math><mi>k</mi><mo>≥</mo><mn>0</mn></math></span>. In this paper, we generalize Rényi's theorem and give a short and elementary proof. Moreover, we show that the distribution of <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>−</mo><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> displays a disjoint form with the arithmetic functions of invariant averages under multiplications. As a consequence, we obtain some ergodic theorems on <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>−</mo><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> that build connections among Rényi's theorem, the prime number theorem, Bergelson-Richter's theorem and Loyd's theorem.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 423-437"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Rankin-Selberg gamma factor of simple supercuspidal representations of the even unitary group for p-adic local fields
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2025.01.015
Philip Barron, Yu Xin
{"title":"On the Rankin-Selberg gamma factor of simple supercuspidal representations of the even unitary group for p-adic local fields","authors":"Philip Barron,&nbsp;Yu Xin","doi":"10.1016/j.jnt.2025.01.015","DOIUrl":"10.1016/j.jnt.2025.01.015","url":null,"abstract":"<div><div>Let <em>π</em> be a simple supercuspidal representation of the quasi-split unramified even unitary group with respect to an unramified quadratic extension <span><math><mi>E</mi><mo>/</mo><mi>F</mi></math></span> of <em>p</em>-adic fields. We compute the Rankin-Selberg gamma factor for rank-1 twists of <em>π</em> by a tamely ramified character of <span><math><msup><mrow><mi>E</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span>. For non-dyadic cases, the gamma factor can also be derived from the fact that endoscopic lift is also simple supercuspidal. For the dyadic case, the result is original. We expect to extend the result on the endoscopic lift to the dyadic case with our computation.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 526-547"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Kodaira types of elliptic curves with potentially good supersingular reduction
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2025.01.008
Haiyang Wang
{"title":"On the Kodaira types of elliptic curves with potentially good supersingular reduction","authors":"Haiyang Wang","doi":"10.1016/j.jnt.2025.01.008","DOIUrl":"10.1016/j.jnt.2025.01.008","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> be a Henselian discrete valuation domain with field of fractions <em>K</em>. Assume that <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> has algebraically closed residue field <em>k</em>. Let <span><math><mi>E</mi><mo>/</mo><mi>K</mi></math></span> be an elliptic curve with additive reduction. The semi-stable reduction theorem asserts that there exists a minimal extension <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> such that the base change <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>/</mo><mi>L</mi></math></span> has semi-stable reduction.</div><div>It is natural to wonder whether specific properties of the semi-stable reduction and of the extension <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> impose restrictions on what types of Kodaira type the special fiber of <span><math><mi>E</mi><mo>/</mo><mi>K</mi></math></span> may have. In this paper we study the restrictions imposed on the reduction type when the extension <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> is wildly ramified of degree 2, and the curve <span><math><mi>E</mi><mo>/</mo><mi>K</mi></math></span> has potentially good supersingular reduction. We also analyze the possible reduction types of two isogenous elliptic curves with these properties.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 283-307"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Sums of four polygonal numbers: Precise formulas
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2025.01.016
Jialin Li, Haowu Wang
{"title":"Sums of four polygonal numbers: Precise formulas","authors":"Jialin Li,&nbsp;Haowu Wang","doi":"10.1016/j.jnt.2025.01.016","DOIUrl":"10.1016/j.jnt.2025.01.016","url":null,"abstract":"<div><div>In this paper we give unified formulas for the numbers of representations of positive integers as sums of four generalized <em>m</em>-gonal numbers, and as restricted sums of four squares under a linear condition, respectively. These formulas are given as <span><math><mi>Z</mi></math></span>-linear combinations of Hurwitz class numbers. As applications, we prove several Zhi-Wei Sun's conjectures. As by-products, we obtain formulas for expressing the Fourier coefficients of <span><math><mi>ϑ</mi><msup><mrow><mo>(</mo><mi>τ</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup></math></span>, <span><math><mi>η</mi><msup><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow><mrow><mn>12</mn></mrow></msup></math></span>, <span><math><mi>η</mi><msup><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup></math></span> and <span><math><mi>η</mi><msup><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow><mrow><mn>8</mn></mrow></msup><mi>η</mi><msup><mrow><mo>(</mo><mn>2</mn><mi>τ</mi><mo>)</mo></mrow><mrow><mn>8</mn></mrow></msup></math></span> in terms of Hurwitz class numbers, respectively. The proof is based on the theory of Jacobi forms.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 407-422"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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