International Journal of Number Theory最新文献

{"title":"The Barnes–Hurwitz zeta cocycle at s = 0 and Ehrhart quasi-polynomials of triangles","authors":"Milton Espinoza","doi":"10.1142/s179304212450057x","DOIUrl":"https://doi.org/10.1142/s179304212450057x","url":null,"abstract":"<p>Following a theorem of Hayes, we give a geometric interpretation of the special value at <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>s</mi><mo>=</mo><mn>0</mn></math></span><span></span> of certain <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn></math></span><span></span>-cocycle on <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">PGL</mtext></mstyle></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℚ</mi><mo stretchy=\"false\">)</mo></math></span><span></span> previously introduced by the author. This work yields three main results: an explicit formula for our cocycle at <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>s</mi><mo>=</mo><mn>0</mn></math></span><span></span>, a generalization and a new proof of Hayes’ theorem, and an elegant summation formula for the zeroth coefficient of the Ehrhart quasi-polynomial of certain triangles in <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>ℝ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span></span>.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"149 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204340","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":"Linear algebra and congruences for MacMahon’s k-rowed plane partitions","authors":"Shi-Chao Chen","doi":"10.1142/s1793042124500702","DOIUrl":"https://doi.org/10.1142/s1793042124500702","url":null,"abstract":"<p>In this paper, we provide an algorithm to detect linear congruences of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, the number of MacMahon’s <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi></math></span><span></span>-rowed plane partitions, and give a quantitative result on the nonexistence of Ramanujan-type congruences of the <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi></math></span><span></span>-rowed plane partition functions. We also show <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo stretchy=\"false\">)</mo></math></span><span></span> that the number of partitions at most <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>m</mi></math></span><span></span> parts always admits linear congruences.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"23 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204255","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":"Fast computation of generalized dedekind sums","authors":"Preston Tranbarger, Jessica Wang","doi":"10.1142/s179304212450060x","DOIUrl":"https://doi.org/10.1142/s179304212450060x","url":null,"abstract":"<p>We construct an algorithm that reduces the complexity for computing generalized Dedekind sums from exponential to polynomial time. We do so by using an efficient word rewriting process in group theory.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"9 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204260","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 Manin–Peyre conjecture for certain multiprojective hypersurfaces","authors":"Xiaodong Zhao","doi":"10.1142/s1793042124500623","DOIUrl":"https://doi.org/10.1142/s1793042124500623","url":null,"abstract":"<p>By the circle method, an asymptotic formula is established for the number of integer points on certain hypersurfaces within multiprojective space. Using Möbius inversion and the modified hyperbola method, we prove the Manin–Peyre conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for certain smooth hypersurfaces in the multiprojective space of sufficiently large dimension.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"17 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204261","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":"Higher Mertens constants for almost primes II","authors":"Jonathan Bayless, Paul Kinlaw, Jared Duker Lichtman","doi":"10.1142/s179304212450088x","DOIUrl":"https://doi.org/10.1142/s179304212450088x","url":null,"abstract":"<p>For <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi><mo>≥</mo><mn>1</mn></math></span><span></span>, let <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">ℛ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span> denote the reciprocal sum up to <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>x</mi></math></span><span></span> of numbers with <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi></math></span><span></span> prime factors, counted with multiplicity. In prior work, the authors obtained estimates for <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">ℛ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, extending Mertens’ second theorem, as well as a finer-scale estimate for <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">ℛ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span> up to <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mo stretchy=\"false\">(</mo><mo>log</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo stretchy=\"false\">−</mo><mi>N</mi></mrow></msup></math></span><span></span> error for any <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>N</mi><mo>&gt;</mo><mn>0</mn></math></span><span></span>. In this paper, we establish the limiting behavior of the higher Mertens constants from the <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">ℛ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span> estimate. We also extend these results to <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">ℛ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, and we comment on the general case <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi><mo>≥</mo><mn>4</mn></math></span><span></span>.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"17 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204259","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":"Variations on a theorem of Capelli","authors":"Pradipto Banerjee","doi":"10.1142/s1793042124500465","DOIUrl":"https://doi.org/10.1142/s1793042124500465","url":null,"abstract":"&lt;p&gt;Elementary irreducibility criteria are established for &lt;span&gt;&lt;math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;ℤ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; is irreducible over &lt;span&gt;&lt;math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;ℚ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; and &lt;span&gt;&lt;math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; is a prime. For instance, our main criterion implies that if &lt;span&gt;&lt;math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; is reducible over &lt;span&gt;&lt;math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;ℚ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, then &lt;span&gt;&lt;math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; divides &lt;span&gt;&lt;math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; modulo &lt;span&gt;&lt;math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;. Among several applications, it is shown that if &lt;span&gt;&lt;math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; has coefficients in &lt;span&gt;&lt;math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mo stretchy=\"false\"&gt;{&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=\"false\"&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, then &lt;span&gt;&lt;math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; is irreducible over &lt;span&gt;&lt;math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;ℚ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; excluding a couple of obvious exceptions. As another application, it is proved that if &lt;span&gt;&lt;math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; and &lt;span&gt;&lt;math altimg=\"eq-00016.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&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;a&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;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"21 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204253","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":"A conjecture of Hegyvári","authors":"Xing-Wang Jiang, Wu-Xia Ma","doi":"10.1142/s1793042124500477","DOIUrl":"https://doi.org/10.1142/s1793042124500477","url":null,"abstract":"<p>For a given sequence <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi></math></span><span></span> of nonnegative integers, let <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>P</mi><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo></math></span><span></span> be the set of all finite subsequence sums of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi></math></span><span></span>. <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi></math></span><span></span> is called complete if <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>P</mi><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo></math></span><span></span> contains all sufficiently large integers. A real number <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>α</mi><mo>&gt;</mo><mn>0</mn></math></span><span></span> is called as an infinite diadical fraction (briefly i.d.f.) if the digit 1 appears infinitely many times in the binary representation of <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>α</mi></math></span><span></span>. Hegyvári conjectured that <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></msub></math></span><span></span> is complete if <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>α</mi></math></span><span></span> or <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>β</mi></math></span><span></span> is i.d.f. and <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>α</mi><mo stretchy=\"false\">/</mo><mi>β</mi><mo>≠</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>l</mi></mrow></msup><mspace width=\"0.25em\"></mspace><mo stretchy=\"false\">(</mo><mi>l</mi><mo>∈</mo><mi>ℤ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, where <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></msub><mo>=</mo><mo stretchy=\"false\">{</mo><mo stretchy=\"false\">[</mo><mi>α</mi><mo stretchy=\"false\">]</mo><mo>,</mo><mo stretchy=\"false\">[</mo><mi>β</mi><mo stretchy=\"false\">]</mo><mo>,</mo><mo>…</mo><mo>,</mo><mo stretchy=\"false\">[</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mi>α</mi><mo stretchy=\"false\">]</mo><mo>,</mo><mo stretchy=\"false\">[</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mi>β</mi><mo stretchy=\"false\">]</mo><mo>,</mo><mo>…</mo><mo stretchy=\"false\">}</mo></math></span><span></span> is a sequence of integers. In this paper, we give a partial result of Hegyvári’s conjecture.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"40 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204335","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":"Values of certain Dirichlet series and higher derivative formulas of trigonometric functions","authors":"Dominic Lanphier, Allen Lin","doi":"10.1142/s1793042124500519","DOIUrl":"https://doi.org/10.1142/s1793042124500519","url":null,"abstract":"<p>We determine new values of certain Dirichlet series and related infinite series. These formulas extend results of several authors. To obtain these results we apply recent expansions of higher derivative formulas of trigonometric functions. We also investigate the transcendentality of values of these series and arithmetic relations of the values of certain related infinite series.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"22 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204257","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 bounded basis with prescribed representation functions","authors":"Fang-Gang Xue","doi":"10.1142/s1793042124500179","DOIUrl":"https://doi.org/10.1142/s1793042124500179","url":null,"abstract":"Let [Formula: see text] be the set of integers and [Formula: see text] the set of positive integers. For a nonempty set [Formula: see text] of integers and any integers [Formula: see text], [Formula: see text] with [Formula: see text], define [Formula: see text] as the number of solutions of [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text] A set [Formula: see text] of integers is defined as a basis of order [Formula: see text] for [Formula: see text] if [Formula: see text] for every integer [Formula: see text]. In 2004, Nešetřil and Serra considered the Erdős–Turán conjecture for a class of bounded bases. In this paper, we generalize the above result and obtain that: for any function [Formula: see text], there exists a bounded basis of order [Formula: see text] for [Formula: see text] such that [Formula: see text] for every integer [Formula: see text].","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"57 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135875133","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":"Infinite families of solutions for A3 + B3 = C3 + D3 and A4 + B4 + C4 + D4 + E4 = F4 in the spirit of Ramanujan","authors":"Archit Agarwal, Meghali Garg","doi":"10.1142/s1793042124500283","DOIUrl":"https://doi.org/10.1142/s1793042124500283","url":null,"abstract":"Ramanujan, in his lost notebook, gave an interesting identity, which generates infinite families of solutions to Euler’s Diophantine equation [Formula: see text]. In this paper, we produce a few infinite families of solutions to the aforementioned Diophantine equation as well as for the Diophantine equation [Formula: see text] in the spirit of Ramanujan.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"57 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135875134","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}