International Journal of Number Theory最新文献

{"title":"Riemann hypothesis for period polynomials for cusp forms on Γ0(N)","authors":"SoYoung Choi","doi":"10.1142/s1793042124500982","DOIUrl":"https://doi.org/10.1142/s1793042124500982","url":null,"abstract":"<p>We prove that for even integer <span><math altimg=\"eq-00003.gif\" display=\"inline\"><mi>k</mi></math></span><span></span>, almost all of zeros of the period polynomial associated to a cusp form of weight <span><math altimg=\"eq-00004.gif\" display=\"inline\"><mi>k</mi></math></span><span></span> on <span><math altimg=\"eq-00005.gif\" display=\"inline\"><msub><mrow><mi mathvariant=\"normal\">Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>N</mi><mo stretchy=\"false\">)</mo></math></span><span></span> are on the circle <span><math altimg=\"eq-00006.gif\" display=\"inline\"><mo>|</mo><mi>z</mi><mo>|</mo><mo>=</mo><mn>1</mn><mo stretchy=\"false\">/</mo><msqrt><mrow><mi>N</mi></mrow></msqrt></math></span><span></span> under some conditions.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"18 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253809","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":"Lehmer-type bounds and counting rational points of bounded heights on Abelian varieties","authors":"Narasimha Kumar, Satyabrat Sahoo","doi":"10.1142/s1793042124501045","DOIUrl":"https://doi.org/10.1142/s1793042124501045","url":null,"abstract":"<p>In this paper, we study Lehmer-type bounds for the Néron–Tate height of <span><math altimg=\"eq-00001.gif\" display=\"inline\"><mover accent=\"true\"><mrow><mi>K</mi></mrow><mo>̄</mo></mover></math></span><span></span>-points on abelian varieties <span><math altimg=\"eq-00002.gif\" display=\"inline\"><mi>A</mi></math></span><span></span> over number fields <span><math altimg=\"eq-00003.gif\" display=\"inline\"><mi>K</mi></math></span><span></span>. Then, we estimate the number of <span><math altimg=\"eq-00004.gif\" display=\"inline\"><mi>K</mi></math></span><span></span>-rational points on <span><math altimg=\"eq-00005.gif\" display=\"inline\"><mi>A</mi></math></span><span></span> with Néron–Tate height <span><math altimg=\"eq-00006.gif\" display=\"inline\"><mo>≤</mo><mo>log</mo><mi>B</mi></math></span><span></span> for <span><math altimg=\"eq-00007.gif\" display=\"inline\"><mi>B</mi><mo>≫</mo><mn>0</mn></math></span><span></span>. This estimate involves a constant <span><math altimg=\"eq-00008.gif\" display=\"inline\"><mi>C</mi></math></span><span></span>, which is not explicit. However, for elliptic curves and the product of elliptic curves over <span><math altimg=\"eq-00009.gif\" display=\"inline\"><mi>K</mi></math></span><span></span>, we make the constant explicitly computable.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"41 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253929","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":"Arithmetic progressions in polynomial orbits","authors":"Mohammad Sadek, Mohamed Wafik, Tuğba Yesin","doi":"10.1142/s1793042124500970","DOIUrl":"https://doi.org/10.1142/s1793042124500970","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\"><mi>f</mi></math></span><span></span> be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit <span><math altimg=\"eq-00002.gif\" display=\"inline\"><msub><mrow><mo>Orb</mo></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo stretchy=\"false\">{</mo><mi>t</mi><mo>,</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo>,</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>,</mo><mo>…</mo><mo stretchy=\"false\">}</mo></math></span><span></span>, where <span><math altimg=\"eq-00003.gif\" display=\"inline\"><mi>t</mi></math></span><span></span> is an integer, using arithmetic progressions each of which contains <span><math altimg=\"eq-00004.gif\" display=\"inline\"><mi>t</mi></math></span><span></span>. Fixing an integer <span><math altimg=\"eq-00005.gif\" display=\"inline\"><mi>k</mi><mo>≥</mo><mn>2</mn></math></span><span></span>, we prove that it is impossible to cover <span><math altimg=\"eq-00006.gif\" display=\"inline\"><msub><mrow><mo>Orb</mo></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo></math></span><span></span> using <span><math altimg=\"eq-00007.gif\" display=\"inline\"><mi>k</mi></math></span><span></span> such arithmetic progressions unless <span><math altimg=\"eq-00008.gif\" display=\"inline\"><msub><mrow><mo>Orb</mo></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is contained in one of these progressions. In fact, we show that the relative density of terms covered by <span><math altimg=\"eq-00009.gif\" display=\"inline\"><mi>k</mi></math></span><span></span> such arithmetic progressions in <span><math altimg=\"eq-00010.gif\" display=\"inline\"><msub><mrow><mo>Orb</mo></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is uniformly bounded from above by a bound that depends solely on <span><math altimg=\"eq-00011.gif\" display=\"inline\"><mi>k</mi></math></span><span></span>. In addition, the latter relative density can be made as close as desired to <span><math altimg=\"eq-00012.gif\" display=\"inline\"><mn>1</mn></math></span><span></span> by an appropriate choice of <span><math altimg=\"eq-00013.gif\" display=\"inline\"><mi>k</mi></math></span><span></span> arithmetic progressions containing <span><math altimg=\"eq-00014.gif\" display=\"inline\"><mi>t</mi></math></span><span></span> if <span><math altimg=\"eq-00015.gif\" display=\"inline\"><mi>k</mi></math></span><span></span> is allowed to be large enough.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"34 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253805","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 mean values for the exponential sum of divisor functions","authors":"Wei Zhang","doi":"10.1142/s1793042124500933","DOIUrl":"https://doi.org/10.1142/s1793042124500933","url":null,"abstract":"<p>In this paper, we study mean values for exponential sums of divisor functions. We improve previous results of [M. Pandey, Moment estimates for the exponential sum with higher divisor functions, <i>C. R. Math. Acad. Sci. Paris</i><b>360</b> (2022) 419–424].</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"44 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146381","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":"Explicit evaluation of triple convolution sums of the divisor functions","authors":"B. Ramakrishnan, Brundaban Sahu, Anup Kumar Singh","doi":"10.1142/s1793042124500544","DOIUrl":"https://doi.org/10.1142/s1793042124500544","url":null,"abstract":"&lt;p&gt;In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums &lt;disp-formula-group&gt;&lt;span&gt;&lt;math altimg=\"eq-00001.gif\" display=\"block\" overflow=\"scroll\"&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&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;d&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;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&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;r&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;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac linethickness=\"0\"&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;l&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;l&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;l&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;mi&gt;ℕ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&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;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&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;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;l&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;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/disp-formula-group&gt; for odd integers &lt;span&gt;&lt;math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&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;r&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;r&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;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=\"0.25em\"&gt;&lt;/mspace&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; and &lt;span&gt;&lt;math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&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;d&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;d&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;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;ℕ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math altimg=\"eq-00004.gif\" display=\"inline\" overf","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"60 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833094","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":"Uniformity of quadratic points","authors":"Tangli Ge","doi":"10.1142/s1793042124500532","DOIUrl":"https://doi.org/10.1142/s1793042124500532","url":null,"abstract":"<p>In this paper, we extend a uniformity result of Dimitrov <i>et al.</i> [Uniformity in Mordell-Lang for curves, <i>Ann. of Math.</i> (<i>2</i>) <b>194</b>(1) (2021) 237–298] to dimension two and use it to get a uniform bound on the cardinality of the set of all quadratic points for non-hyperelliptic non-bielliptic curves which only depend on the Mordell–Weil rank, the genus of the curve and the degree of the number field.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"56 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833108","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 𝔭-primary uniform boundedness conjecture for Drinfeld modules","authors":"Shun Ishii","doi":"10.1142/s1793042124500611","DOIUrl":"https://doi.org/10.1142/s1793042124500611","url":null,"abstract":"<p>In this paper, we study a Drinfeld module analogue of the Uniform Boundedness Conjecture on the torsion of abelian varieties. As a result, we prove the <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝔭</mi></math></span><span></span>-primary Uniform Boundedness Conjecture for one-dimensional families of Drinfeld modules of arbitrary rank, which extends a result of Poonen. This result can be regarded as a Drinfeld module analogue of the Cadoret–Tamagawa’s result on the <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-primary Uniform Boundedness Conjecture for one-dimensional families of abelian varieties.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"18 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833113","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":"Near-miss identities and spinor genus classification of ternary quadratic forms with congruence conditions","authors":"Kush Singhal","doi":"10.1142/s1793042124500507","DOIUrl":"https://doi.org/10.1142/s1793042124500507","url":null,"abstract":"<p>In this paper, near-miss identities for the number of representations of some integral ternary quadratic forms with congruence conditions are found and proven. The genus and spinor genus of the corresponding lattice cosets are then classified. Finally, a complete genus and spinor genus classification for all conductor 2 lattice cosets of 2-adically unimodular lattices is given.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"25 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833165","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":"Dense clusters of zeros near the zero-free region of ζ(s)","authors":"William D. Banks","doi":"10.1142/s1793042124500520","DOIUrl":"https://doi.org/10.1142/s1793042124500520","url":null,"abstract":"<p>The methods of Korobov and Vinogradov produce a zero-free region for the Riemann zeta function <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>ζ</mi><mo stretchy=\"false\">(</mo><mi>s</mi><mo stretchy=\"false\">)</mo></math></span><span></span> of the form <disp-formula-group><span><math altimg=\"eq-00004.gif\" display=\"block\" overflow=\"scroll\"><mrow><mi>σ</mi><mo>&gt;</mo><mn>1</mn><mo stretchy=\"false\">−</mo><mfrac><mrow><mi>c</mi></mrow><mrow><msup><mrow><mo stretchy=\"false\">(</mo><mo>log</mo><mi>τ</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mn>2</mn><mo stretchy=\"false\">/</mo><mn>3</mn></mrow></msup><msup><mrow><mo stretchy=\"false\">(</mo><mo>log</mo><mo>log</mo><mi>τ</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mn>1</mn><mo stretchy=\"false\">/</mo><mn>3</mn></mrow></msup></mrow></mfrac><mspace width=\"1em\"></mspace><mo stretchy=\"false\">(</mo><mi>τ</mi><mo>≔</mo><mo stretchy=\"false\">|</mo><mi>t</mi><mo stretchy=\"false\">|</mo><mo stretchy=\"false\">+</mo><mn>1</mn><mn>0</mn><mn>0</mn><mo stretchy=\"false\">)</mo><mo>.</mo></mrow></math></span><span></span></disp-formula-group> For many decades, the general shape of the zero-free region has not changed (although explicit known values for <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>c</mi></math></span><span></span> have improved over the years). In this paper, we show that if the zero-free region <i>cannot</i> be widened substantially, then there exist infinitely many distinct dense clusters of zeros of <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>ζ</mi><mo stretchy=\"false\">(</mo><mi>s</mi><mo stretchy=\"false\">)</mo></math></span><span></span> lying close to the edge of the zero-free region. Our proof provides specific information about the location of these clusters and the number of zeros contained in them. To prove the result, we introduce and apply a variant of the original method of de la Vallée Poussin combined with ideas of Turán to control the real parts of power sums. We also prove similar results for <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>L</mi></math></span><span></span>-functions associated to <i>nonquadratic</i> Dirichlet characters <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>χ</mi></math></span><span></span> modulo <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>q</mi><mo>≥</mo><mn>2</mn></math></span><span></span>.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"51 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800385","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 note on logarithmic equidistribution","authors":"Gerold Schefer","doi":"10.1142/s1793042124500647","DOIUrl":"https://doi.org/10.1142/s1793042124500647","url":null,"abstract":"<p>For every algebraic number <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>κ</mi></math></span><span></span> on the unit circle which is not a root of unity we prove the existence of a strict sequence of algebraic numbers whose height tends to zero, such that the averages of the evaluation of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>f</mi></mrow><mrow><mi>κ</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>log</mo><mspace width=\"-.17em\"></mspace><mo>|</mo><mi>z</mi><mo stretchy=\"false\">−</mo><mi>κ</mi><mo>|</mo></math></span><span></span> at the conjugates are essentially bounded from above by <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">−</mo><mi>h</mi><mo stretchy=\"false\">(</mo><mi>κ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. This completes a characterization on functions <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>f</mi></mrow><mrow><mi>κ</mi></mrow></msub></math></span><span></span> initiated by Autissier and Baker–Masser, who cover the cases <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>κ</mi><mo>=</mo><mn>2</mn></math></span><span></span> and <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mo>|</mo><mi>κ</mi><mo>|</mo><mo>≠</mo><mn>1</mn></math></span><span></span>, respectively. Using the same ideas we also prove analogues in the <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-adic setting.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"27 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800310","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}