{"title":"Upper bounds on large deviations of Dirichlet L-functions in the q-aspect","authors":"Louis-Pierre Arguin, Nathan Creighton","doi":"10.1016/j.jnt.2025.01.009","DOIUrl":"10.1016/j.jnt.2025.01.009","url":null,"abstract":"<div><div>We prove a result on the large deviations of the central values of even primitive Dirichlet <em>L</em>-functions with a given modulus. For <span><math><mi>V</mi><mo>∼</mo><mi>α</mi><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi></math></span> with <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></math></span>, we show that<span><span><span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>φ</mi><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mfrac><mi>#</mi><mrow><mo>{</mo><mi>χ</mi><mtext> even, primitive mod </mtext><mi>q</mi><mo>:</mo><mi>log</mi><mo></mo><mrow><mo>|</mo><mi>L</mi><mrow><mo>(</mo><mi>χ</mi><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>|</mo></mrow><mo>></mo><mi>V</mi><mo>}</mo></mrow><mspace></mspace><mo>≪</mo><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mfrac><mrow><msup><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi></mrow></mfrac></mrow></msup></mrow><mrow><msqrt><mrow><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi></mrow></msqrt></mrow></mfrac><mo>.</mo></math></span></span></span> This yields the sharp upper bound for the fractional moments of central values of Dirichlet <em>L</em>-functions proved by Gao, upon noting that the number of even, primitive characters with modulus <em>q</em> is <span><math><mfrac><mrow><mi>φ</mi><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. The proof is an adaptation to the <em>q</em>-aspect of the recursive scheme developed by Arguin, Bourgade and Radziwiłł for the local maxima of the Riemann zeta function, and applied by Arguin and Bailey to the large deviations in the <em>t</em>-aspect. We go further and get bounds on the case where <span><math><mi>V</mi><mo>=</mo><mi>o</mi><mo>(</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>q</mi><mo>)</mo></math></span>. These bounds are not expected to be sharp, but the discrepancy from the Central Limit Theorem estimate grows very slowly with <em>q</em>. The method involves a formula for the twisted mollified second moment of central values of Dirichlet <em>L</em>-functions, building on the work of Iwaniec and Sarnak.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"273 ","pages":"Pages 96-158"},"PeriodicalIF":0.6,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511909","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}
{"title":"Square patterns in dynamical orbits","authors":"Vefa Goksel , Giacomo Micheli","doi":"10.1016/j.jnt.2024.12.004","DOIUrl":"10.1016/j.jnt.2024.12.004","url":null,"abstract":"<div><div>Let <em>q</em> be an odd prime power. Let <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> be a polynomial having degree at least 2, <span><math><mi>a</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, and denote by <span><math><msup><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> the <em>n</em>-th iteration of <em>f</em>. Let <em>χ</em> be the quadratic character of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, and <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>)</mo></math></span> the forward orbit of <em>a</em> under iteration by <em>f</em>. Suppose that the sequence <span><math><msub><mrow><mo>(</mo><mi>χ</mi><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>a</mi><mo>)</mo><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></math></span> is periodic, and <em>m</em> is its period. Assuming a mild and generic condition on <em>f</em>, we show that, up to a constant depending on <em>d</em>, <em>m</em> can be bounded from below by <span><math><mo>|</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>)</mo><mo>|</mo><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mfrac><mrow><mn>2</mn><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mo>(</mo><mi>d</mi><mo>)</mo><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mo>(</mo><mi>d</mi><mo>)</mo><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></msup></math></span> as <em>q</em> grows. More informally, we prove that the period of the appearance of squares in an orbit of an element provides an upper bound for the size of the orbit itself. Using a similar method, we can also prove that, up to a constant depending on <em>d</em>, we cannot have more than <span><math><msup><mrow><mi>q</mi></mrow><mrow><mfrac><mrow><mn>2</mn><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mo>(</mo><mi>d</mi><mo>)</mo><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mo>(</mo><mi>d</mi><mo>)</mo><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></msup></math></span> consecutive squares or non-squares in the forward orbit of <em>a</em>. In addition, using geometric tools from global function field theory such as abc theorem, we provide a classification of all polynomials for which our generic condition does not hold, making the results effective. Interestingly enough, our condition is purely geometrical, while our final results are completely arithmetical. As a corollary, this paper removes most of the hypothesis of (Ostafe, Shparlinski. Proceedings of the American Mathematical Society 138.8 (2010)), most notably extending the results to an","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"272 ","pages":"Pages 129-146"},"PeriodicalIF":0.6,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143512223","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}
{"title":"An asymptotic formula involving the triple divisor function","authors":"Guangwei Hu , Chenran Xu","doi":"10.1016/j.jnt.2025.01.011","DOIUrl":"10.1016/j.jnt.2025.01.011","url":null,"abstract":"<div><div>Suppose <span><math><msub><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denotes the classical triple divisor function. Let <span><math><mi>Q</mi><mo>(</mo><mrow><mi>x</mi><mo>)</mo></mrow></math></span> be a positive definite integral quadratic form, and <span><math><mi>r</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> denote the number of representations of <em>n</em> by the quadratic form <em>Q</em>. In this paper, we will establish an asymptotic formula of the summation<span><span><span><math><munder><mo>∑</mo><mrow><mi>n</mi><mo>≤</mo><mi>X</mi></mrow></munder><msub><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>+</mo><mi>h</mi><mo>)</mo><mi>r</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>Q</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <em>h</em> is a positive integer satisfying <span><math><mi>h</mi><mo>≤</mo><mi>H</mi><mo>≪</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>ε</mi></mrow></msup></math></span>. Our result breaks through the trivial bound of the above summation and obtains the power saving in <em>O</em>-term.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 328-347"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445484","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}
{"title":"Some improvements on the Davenport-Heilbronn method","authors":"Konstantinos Kydoniatis","doi":"10.1016/j.jnt.2025.01.013","DOIUrl":"10.1016/j.jnt.2025.01.013","url":null,"abstract":"<div><div>Let <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, <span><math><mi>s</mi><mo>≥</mo><mo>⌈</mo><mi>k</mi><mo>(</mo><mi>log</mi><mo></mo><mi>k</mi><mo>+</mo><mn>4.20032</mn><mo>)</mo><mo>⌉</mo></math></span>, and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mi>ω</mi><mo>∈</mo><mi>R</mi></math></span>. Assume that the <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> are non-zero, not all in rational ratio, and not all of the same sign in the case that <em>k</em> is even. Then, for any <span><math><mi>ϵ</mi><mo>></mo><mn>0</mn></math></span>, the inequality<span><span><span><math><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>s</mi></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>+</mo><mi>ω</mi><mo>|</mo><mo><</mo><mi>ϵ</mi></math></span></span></span> has <span><math><mo>≫</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>s</mi><mo>−</mo><mi>k</mi></mrow></msup></math></span> integer solutions with <span><math><mo>|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>≤</mo><mi>P</mi></math></span>. Moreover the asymptotic formula for the number of smooth solutions is established assuming the same conditions hold.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"272 ","pages":"Pages 1-17"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474666","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}
{"title":"Heights of rational points on Mordell curves","authors":"Alan Zhao","doi":"10.1016/j.jnt.2025.01.012","DOIUrl":"10.1016/j.jnt.2025.01.012","url":null,"abstract":"<div><div>We conjecture a lower bound for the minimal canonical height of non-torsion rational points on a natural density 1 subset of the sextic twist family of Mordell curves. We then establish a lower bound that yields a partial result towards this conjecture.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"272 ","pages":"Pages 18-33"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474667","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}
{"title":"Negative first moment of quadratic twists of L-functions","authors":"Peng Gao , 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}
{"title":"Approximation of L-functions associated to Hecke cusp eigenforms","authors":"An Huang, 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}
{"title":"On some unramified families of motivic Euler sums","authors":"Ce Xu , 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}
{"title":"Kummer theory for function fields","authors":"Félix Baril Boudreau, 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}
{"title":"On a twisted Jacquet module of GL(2n) over a finite field","authors":"Kumar Balasubramanian , Abhishek Dangodara , 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}
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