Journal of Number Theory最新文献

{"title":"General multiple Dirichlet series from perverse sheaves","authors":"Will Sawin","doi":"10.1016/j.jnt.2024.03.020","DOIUrl":"10.1016/j.jnt.2024.03.020","url":null,"abstract":"<div><p>We give an axiomatic characterization of multiple Dirichlet series over the function field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math></span>, generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the coefficients, formalizes an observation of Chinta. The existence of multiple Dirichlet series satisfying these axioms is proved by exhibiting the coefficients as trace functions of explicit perverse sheaves and using properties of perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 408-453"},"PeriodicalIF":0.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140768352","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":"Torsion points of elliptic curves over multi-quadratic number fields","authors":"Koji Matsuda","doi":"10.1016/j.jnt.2024.03.018","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.03.018","url":null,"abstract":"<div><p>We compute the Mordell–Weil groups of the modular Jacobian varieties of hyperelliptic modular curves <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>,</mo><mi>M</mi><mi>N</mi><mo>)</mo></math></span> over every composite field of some quadratic number fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 28-43"},"PeriodicalIF":0.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24000933/pdfft?md5=f88774ff4e0aef9e748abddb14237f52&pid=1-s2.0-S0022314X24000933-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140639190","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":"A note on the two variable Artin's conjecture","authors":"S.G. Hazra ,&nbsp;M. Ram Murty ,&nbsp;J. Sivaraman","doi":"10.1016/j.jnt.2024.03.008","DOIUrl":"10.1016/j.jnt.2024.03.008","url":null,"abstract":"&lt;div&gt;&lt;p&gt;In 1927, Artin conjectured that any integer &lt;em&gt;a&lt;/em&gt; which is not −1 or a perfect square is a primitive root for a positive density of primes &lt;em&gt;p&lt;/em&gt;. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers &lt;em&gt;a&lt;/em&gt; and &lt;em&gt;b&lt;/em&gt;, the set&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;⩽&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mtext&gt; prime, &lt;/mtext&gt;&lt;mphantom&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mphantom&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;#&lt;/mi&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;⩽&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mtext&gt; prime, &lt;/mtext&gt;&lt;mphantom&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mphantom&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;mo&gt;≫&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; such that&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mro","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 161-185"},"PeriodicalIF":0.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140767901","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 the modulo p zeros of modular forms congruent to theta series","authors":"Berend Ringeling","doi":"10.1016/j.jnt.2024.03.019","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.03.019","url":null,"abstract":"<div><p>For a prime <em>p</em> larger than 7, the Eisenstein series of weight <span><math><mi>p</mi><mo>−</mo><mn>1</mn></math></span> has some remarkable congruence properties modulo <em>p</em>. Those imply, for example, that the <em>j</em>-invariants of its zeros (which are known to be real algebraic numbers in the interval <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1728</mn><mo>]</mo></math></span>), are at most quadratic over the field with <em>p</em> elements and are congruent modulo <em>p</em> to the zeros of a certain truncated hypergeometric series. In this paper we introduce “theta modular forms” of weight <span><math><mi>k</mi><mo>≥</mo><mn>4</mn></math></span> for the full modular group as the modular forms for which the first <span><math><mi>dim</mi><mo>⁡</mo><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the <em>j</em>-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo <em>p</em> all in the ground field with <em>p</em> elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with <em>p</em> elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 386-407"},"PeriodicalIF":0.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24000921/pdfft?md5=039b6bb5c8fac784d3fa69a2ccefbe61&pid=1-s2.0-S0022314X24000921-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140823699","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":"Patterns of primes in joint Sato–Tate distributions","authors":"A. Anas Chentouf ,&nbsp;Catherine H. Cossaboom ,&nbsp;Samuel E. Goldberg ,&nbsp;Jack B. Miller","doi":"10.1016/j.jnt.2024.03.009","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.03.009","url":null,"abstract":"<div><p>For <span><math><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></span>, let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn><mi>π</mi><mi>i</mi><mi>n</mi><mi>z</mi></mrow></msup></math></span> be a holomorphic, non-CM cuspidal newform of even weight <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≥</mo><mn>2</mn></math></span> with trivial nebentypus. For each prime <em>p</em>, let <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>(</mo><mi>p</mi><mo>)</mo><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>]</mo></math></span> be the angle such that <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>(</mo><mi>p</mi><mo>)</mo><mo>=</mo><mn>2</mn><msup><mrow><mi>p</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></mrow></msup><mi>cos</mi><mo>⁡</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>(</mo><mi>p</mi><mo>)</mo></math></span>. The now-proven Sato–Tate conjecture states that the angles <span><math><mo>(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>(</mo><mi>p</mi><mo>)</mo><mo>)</mo></math></span> equidistribute with respect to the measure <span><math><mi>d</mi><msub><mrow><mi>μ</mi></mrow><mrow><mi>S</mi><mi>T</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>π</mi></mrow></mfrac><msup><mrow><mi>sin</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mi>θ</mi><mspace></mspace><mi>d</mi><mi>θ</mi></math></span>. We show that, if <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is not a character twist of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, then for subintervals <span><math><msub><mrow><mi>I</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊆</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>]</mo></math></span>, there exist infinitely many bounded gaps between the primes <em>p</em> such that <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>p</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>p</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. We also prove a common generalization of the bounded gaps with the Green–Tao theorem.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"263 ","pages":"Pages 297-334"},"PeriodicalIF":0.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141249923","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":"Riemann zeta functions for Krull monoids","authors":"Felix Gotti ,&nbsp;Ulrich Krause","doi":"10.1016/j.jnt.2024.03.001","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.03.001","url":null,"abstract":"<div><p>The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with torsion class groups. In doing so, the Decay Theorem is fundamental as it allows us to use strong atoms instead of primes to obtain a weaker version of the Fundamental Theorem of Arithmetic in the more general setting of Krull monoids with torsion class groups. Several related examples are exhibited throughout the paper, in particular, algebraic number fields for which the generalized Riemann zeta function specializes to the Dedekind zeta function.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 134-160"},"PeriodicalIF":0.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140807970","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":"The circle method and shifted convolution sums involving the divisor function","authors":"Guangwei Hu,&nbsp;Huixue Lao","doi":"10.1016/j.jnt.2024.03.007","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.03.007","url":null,"abstract":"<div><p>Let <span><math><mi>Q</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> be a positive definite integral quadratic form with the determinant <em>D</em> being squarefree, 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 apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function <span><math><mi>d</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and Fourier coefficients <span><math><mi>r</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span>. With more efforts, our method should have a number of applications for other multiplicative functions.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 1-27"},"PeriodicalIF":0.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140639189","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":"Comparison of integral structures on the space of modular forms of full level N","authors":"Anthony Kling","doi":"10.1016/j.jnt.2024.03.015","DOIUrl":"10.1016/j.jnt.2024.03.015","url":null,"abstract":"<div><p>Let <span><math><mi>N</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><mi>r</mi><mo>≥</mo><mn>1</mn></math></span> be integers and <span><math><mi>p</mi><mo>≥</mo><mn>2</mn></math></span> be a prime such that <span><math><mi>p</mi><mo>∤</mo><mi>N</mi></math></span>. One can consider two different integral structures on the space of modular forms over <span><math><mi>Q</mi></math></span>, one coming from arithmetic via <em>q</em>-expansions, the other coming from geometry via integral models of modular curves. Both structures are stable under the Hecke operators; furthermore, their quotient is finite torsion. Our goal is to investigate the exponent of the annihilator of the quotient. We will apply methods due to Brian Conrad to the situation of modular forms of even weight and level <span><math><mi>Γ</mi><mo>(</mo><mi>N</mi><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></math></span> over <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mi>N</mi><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub><mo>)</mo></math></span> to obtain an upper bound for the exponent. We also use Klein forms to construct explicit modular forms of level <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> whenever <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>&gt;</mo><mn>3</mn></math></span>, allowing us to compute a lower bound which agrees with the upper bound. Hence we compute the exponent precisely.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 222-300"},"PeriodicalIF":0.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140774680","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":"Numbers of the form k + f(k)","authors":"Mikhail R. Gabdullin ,&nbsp;Vitalii V. Iudelevich ,&nbsp;Florian Luca","doi":"10.1016/j.jnt.2024.03.010","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.03.010","url":null,"abstract":"<div><p>For a function <span><math><mi>f</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>N</mi></math></span>, let<span><span><span><math><msubsup><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>n</mi><mo>⩽</mo><mi>x</mi><mo>:</mo><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mi>f</mi><mo>(</mo><mi>k</mi><mo>)</mo><mtext> for some </mtext><mi>k</mi><mo>}</mo><mo>.</mo></math></span></span></span> Let <span><math><mi>τ</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>d</mi><mo>|</mo><mi>n</mi></mrow></msub><mn>1</mn></math></span> be the divisor function, <span><math><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>p</mi><mo>|</mo><mi>n</mi></mrow></msub><mn>1</mn></math></span> be the prime divisor function, and <span><math><mi>φ</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>#</mi><mo>{</mo><mn>1</mn><mo>⩽</mo><mi>k</mi><mo>⩽</mo><mi>n</mi><mo>:</mo><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>}</mo></math></span> be Euler's totient function. We show that<span><span><span><math><mo>(</mo><mn>1</mn><mo>)</mo><mspace></mspace><mi>x</mi><mo>≪</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>ω</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mo>(</mo><mn>2</mn><mo>)</mo><mspace></mspace><mi>x</mi><mo>≪</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>τ</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo><mo>⩽</mo><mn>0.94</mn><mi>x</mi><mo>,</mo><mo>(</mo><mn>3</mn><mo>)</mo><mspace></mspace><mi>x</mi><mo>≪</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>φ</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo><mo>⩽</mo><mn>0.93</mn><mi>x</mi><mo>.</mo></math></span></span></span></p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 58-85"},"PeriodicalIF":0.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140649397","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":"Cullen numbers and Woodall numbers in generalized Fibonacci sequences","authors":"Attila Bérczes ,&nbsp;István Pink ,&nbsp;Paul Thomas Young","doi":"10.1016/j.jnt.2024.03.006","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.03.006","url":null,"abstract":"<div><p>Recently Bilu, Marques and Togbé <span>[4]</span> gave a general effective finiteness result on the equation<span><span><span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo></math></span></span></span> where <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup></math></span> denotes the <em>k</em>-generalized Fibonacci-sequence and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> the sequence of Cullen numbers, by giving explicit absolute bounds for <span><math><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi></math></span>. However, the authors in <span>[4]</span> explained that their bounds were too large to use Dujella-Pethő reduction to completely solve the equation in question. In the present paper, using the bounds established by Bilu, Marques and Togbé in <span>[4]</span> and a different approach based on 2-adic analysis, we completely solve this equation. Further, using the same approach we also solve the corresponding equation for Woodall numbers.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 86-102"},"PeriodicalIF":0.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140649398","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}