Pub Date : 2024-04-18DOI: 10.1142/s1793042124500556
Debanjana Kundu, Anwesh Ray
We study the average behavior of the Iwasawa invariants for Selmer groups of elliptic curves. These results lie at the intersection of arithmetic statistics and Iwasawa theory. We obtain lower bounds for the density of rational elliptic curves with prescribed Iwasawa invariants.
{"title":"Statistics for Iwasawa invariants of elliptic curves, II","authors":"Debanjana Kundu, Anwesh Ray","doi":"10.1142/s1793042124500556","DOIUrl":"https://doi.org/10.1142/s1793042124500556","url":null,"abstract":"<p>We study the average behavior of the Iwasawa invariants for Selmer groups of elliptic curves. These results lie at the intersection of arithmetic statistics and Iwasawa theory. We obtain lower bounds for the density of rational elliptic curves with prescribed Iwasawa invariants.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"39 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140625163","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}
Pub Date : 2024-04-10DOI: 10.1142/s1793042124500854
Babita, Mohit Tripathi, Lalit Vaishya
Let be a normalized Hecke eigenform of weight for the full modular group . In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg -functions , supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant We improve previous results in the case when the reduced form is given by
设 f 是全模态群 SL2(ℤ) 权重为 k 的归一化赫克特征形式。在本文中，我们得到了与 Rankin-Selberg L 函数 R(s,f×f) 的傅里叶系数相关的一般除数函数的高阶矩的渐近值，R(s,f×f) 在整数处由固定判别式 D<0 的原始积分正定二元二次方程形式（还原形式）表示。当还原形式为𝒬(x1,x2)=x12+x22 时，我们改进了以前的结果。
{"title":"Oscillations of Fourier coefficients of product of L-functions at integers in a sparse set","authors":"Babita, Mohit Tripathi, Lalit Vaishya","doi":"10.1142/s1793042124500854","DOIUrl":"https://doi.org/10.1142/s1793042124500854","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi></math></span><span></span> be a normalized Hecke eigenform of weight <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi></math></span><span></span> for the full modular group <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℤ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>L</mi></math></span><span></span>-functions <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>R</mi><mo stretchy=\"false\">(</mo><mi>s</mi><mo>,</mo><mi>f</mi><mo stretchy=\"false\">×</mo><mi>f</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>D</mi><mo><</mo><mn>0</mn><mo>.</mo></math></span><span></span> We improve previous results in the case when the reduced form is given by <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"cal\">𝒬</mi><mo stretchy=\"false\">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">)</mo><mo>=</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo stretchy=\"false\">+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>.</mo></math></span><span></span></p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"79 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599024","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}
Pub Date : 2024-04-10DOI: 10.1142/s1793042124500842
Jiankang Wang, Zhefeng Xu
<p>For an odd prime <span><math altimg="eq-00001.gif" display="inline" overflow="scroll"><mi>p</mi></math></span><span></span>, let <span><math altimg="eq-00002.gif" display="inline" overflow="scroll"><msub><mrow><mi>𝔽</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span><span></span> be the finite field of <span><math altimg="eq-00003.gif" display="inline" overflow="scroll"><mi>p</mi></math></span><span></span> elements. The main purpose of this paper is to establish new results on gaps between the elements of multiplicative subgroups of finite fields. For any <span><math altimg="eq-00004.gif" display="inline" overflow="scroll"><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><msubsup><mrow><mi>𝔽</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo stretchy="false">∗</mo></mrow></msubsup></math></span><span></span>, we also obtain new upper bounds of the following double character sum <disp-formula-group><span><math altimg="eq-00005.gif" display="block" overflow="scroll"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></msub><mo stretchy="false">(</mo><mi>χ</mi><mo>,</mo><msub><mrow><mi mathvariant="cal">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="cal">ℋ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant="cal">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant="cal">ℋ</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></munder><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">+</mo><mi>b</mi><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">+</mo><mi>c</mi><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></math></span><span></span></disp-formula-group> and a triple character sum <disp-formula-group><span><math altimg="eq-00006.gif" display="block" overflow="scroll"><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>χ</mi></mrow></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><msub><mrow><mi mathvariant="cal">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="cal">ℋ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi mathvariant="cal">𝒩</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>x</mi><mo>∈</mo><mi mathvariant="cal">𝒩</mi></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant="cal">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant="cal">ℋ</mi>
{"title":"Double and triple character sums and gaps between the elements of subgroups of finite fields","authors":"Jiankang Wang, Zhefeng Xu","doi":"10.1142/s1793042124500842","DOIUrl":"https://doi.org/10.1142/s1793042124500842","url":null,"abstract":"<p>For an odd prime <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>, let <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>𝔽</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span><span></span> be the finite field of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span> elements. The main purpose of this paper is to establish new results on gaps between the elements of multiplicative subgroups of finite fields. For any <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><msubsup><mrow><mi>𝔽</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msubsup></math></span><span></span>, we also obtain new upper bounds of the following double character sum <disp-formula-group><span><math altimg=\"eq-00005.gif\" display=\"block\" overflow=\"scroll\"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>χ</mi><mo>,</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">)</mo><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></munder><mi>χ</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">+</mo><mi>b</mi><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy=\"false\">+</mo><mi>c</mi><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">)</mo></mrow></math></span><span></span></disp-formula-group> and a triple character sum <disp-formula-group><span><math altimg=\"eq-00006.gif\" display=\"block\" overflow=\"scroll\"><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>χ</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi mathvariant=\"cal\">𝒩</mi><mo stretchy=\"false\">)</mo><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>x</mi><mo>∈</mo><mi mathvariant=\"cal\">𝒩</mi></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"14 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599021","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}
Pub Date : 2024-04-06DOI: 10.1142/s1793042124500672
Veekesh Kumar
<p>Let <span><math altimg="eq-00001.gif" display="inline" overflow="scroll"><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msup><mo stretchy="false">+</mo><mo>⋯</mo><mo stretchy="false">+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>ℚ</mi><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo></math></span><span></span>, <span><math altimg="eq-00002.gif" display="inline" overflow="scroll"><msub><mrow><mi>a</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>></mo><mn>0</mn></math></span><span></span>, be a polynomial of degree <span><math altimg="eq-00003.gif" display="inline" overflow="scroll"><mi>d</mi><mo>≥</mo><mn>2</mn></math></span><span></span>. Let <span><math altimg="eq-00004.gif" display="inline" overflow="scroll"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math></span><span></span> be a sequence of integers satisfying <disp-formula-group><span><math altimg="eq-00005.gif" display="block" overflow="scroll"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo stretchy="false">+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mspace width="1em"></mspace><mstyle><mtext>for all </mtext></mstyle><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mspace width="1em"></mspace><mstyle><mtext>and</mtext></mstyle><mspace width="1em"></mspace><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><mi>∞</mi><mspace width="1em"></mspace><mstyle><mtext>as </mtext></mstyle><mi>n</mi><mo>→</mo><mi>∞</mi><mo>.</mo></mrow></math></span><span></span></disp-formula-group></p><p>Set <span><math altimg="eq-00006.gif" display="inline" overflow="scroll"><mi>α</mi><mo>:</mo><mo>=</mo><msub><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><msup><mrow><mi>d</mi></mrow><mrow><mo stretchy="false">−</mo><mi>n</mi></mrow></msup></mrow></msubsup></math></span><span></span>. Then, under the assumption <span><math altimg="eq-00007.gif" display="inline" overflow="scroll"><msubsup><mrow><mi>a</mi></mrow><mrow><mi>d</mi></mrow><mrow><mn>1</mn><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>∈</mo><mi>ℚ</mi></math></span><span></span>, in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, <i>Ramanujan J.</i><b>57</b> (2022) 569–581], either <span><math altimg="eq-00008.gif" display="inline" overflow="scroll"><mi>α</mi></math></span><span></span> is transcendental or <span><math altimg="eq-00009.
设 P(x):=adxd+⋯+a0∈ℚ[x]，ad>0，为阶数 d≥2 的多项式。设 (xn) 为一整数序列，满足 xn+1=P(xn)for all n=0,1,2,...，且 xn→∞as n→∞.设 α:=limn→∞xnd-n。那么，在 ad1/(d-1)∈ℚ的假设下，在最近的一个结果 [A. Dubickas, Transcendency of n.Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J.57 (2022) 569-581] 的最新结果中，要么 α 是超越的，要么 α 可以是一个整数或二次皮索单元，α-1 是它在ℚ 上的共轭。在本文中，我们在不假设 ad1/(d-1) 在 ℚ 中的情况下研究了这种 α 的性质，并证明了要么 α 是超越数，要么 αh 是 Pisot 数，而 h 是数域 ℚα,ad-1d-1 的伽罗瓦闭的扭转子群的阶。本文提出的其他结果研究了在 (n,q1,...,qk)∈ℕ×(K×)k 中不等式 ||q1α1n+⋯+qkαkn+β||<𝜃n 的解，考虑了 β 是有理数还是无理数。这里，K 代表一个数域，𝜃∈(0,1)。符号 ||x|| 表示 x 与其在 ℤ 中最接近的整数之间的距离。
{"title":"The transcendence of growth constants associated with polynomial recursions","authors":"Veekesh Kumar","doi":"10.1142/s1793042124500672","DOIUrl":"https://doi.org/10.1142/s1793042124500672","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>P</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>:</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msup><mo stretchy=\"false\">+</mo><mo>⋯</mo><mo stretchy=\"false\">+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>ℚ</mi><mo stretchy=\"false\">[</mo><mi>x</mi><mo stretchy=\"false\">]</mo></math></span><span></span>, <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>a</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>></mo><mn>0</mn></math></span><span></span>, be a polynomial of degree <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>d</mi><mo>≥</mo><mn>2</mn></math></span><span></span>. Let <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> be a sequence of integers satisfying <disp-formula-group><span><math altimg=\"eq-00005.gif\" display=\"block\" overflow=\"scroll\"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo stretchy=\"false\">+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>P</mi><mo stretchy=\"false\">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo><mspace width=\"1em\"></mspace><mstyle><mtext>for all </mtext></mstyle><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mspace width=\"1em\"></mspace><mstyle><mtext>and</mtext></mstyle><mspace width=\"1em\"></mspace><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><mi>∞</mi><mspace width=\"1em\"></mspace><mstyle><mtext>as </mtext></mstyle><mi>n</mi><mo>→</mo><mi>∞</mi><mo>.</mo></mrow></math></span><span></span></disp-formula-group></p><p>Set <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>α</mi><mo>:</mo><mo>=</mo><msub><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><msup><mrow><mi>d</mi></mrow><mrow><mo stretchy=\"false\">−</mo><mi>n</mi></mrow></msup></mrow></msubsup></math></span><span></span>. Then, under the assumption <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mi>a</mi></mrow><mrow><mi>d</mi></mrow><mrow><mn>1</mn><mo stretchy=\"false\">/</mo><mo stretchy=\"false\">(</mo><mi>d</mi><mo stretchy=\"false\">−</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow></msubsup><mo>∈</mo><mi>ℚ</mi></math></span><span></span>, in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, <i>Ramanujan J.</i><b>57</b> (2022) 569–581], either <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>α</mi></math></span><span></span> is transcendental or <span><math altimg=\"eq-00009.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"300 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599455","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}
Pub Date : 2024-04-06DOI: 10.1142/s1793042124500490
Lei Shi, Qiming Yan
In this paper, motivated by Nochka weights and the replacing hypersurfaces technique, we give an improvement of Schmidt’s subspace type theorem for hypersurfaces which are located in subgeneral position.
{"title":"A note on Schmidt’s subspace type theorems for hypersurfaces in subgeneral position","authors":"Lei Shi, Qiming Yan","doi":"10.1142/s1793042124500490","DOIUrl":"https://doi.org/10.1142/s1793042124500490","url":null,"abstract":"<p>In this paper, motivated by Nochka weights and the replacing hypersurfaces technique, we give an improvement of Schmidt’s subspace type theorem for hypersurfaces which are located in subgeneral position.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"32 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599264","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 new upper bound on Ruzsa’s numbers on the Erdős–Turán conjecture","authors":"Yuchen Ding, Lilu Zhao","doi":"10.1142/s179304212450074x","DOIUrl":"https://doi.org/10.1142/s179304212450074x","url":null,"abstract":"<p>In this paper, we show that the Ruzsa number <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span><span></span> is bounded by <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn><mn>9</mn><mn>2</mn></math></span><span></span> for any positive integer <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>m</mi></math></span><span></span>, which improves the prior bound <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>≤</mo><mn>2</mn><mn>8</mn><mn>8</mn></math></span><span></span> given by Chen in 2008.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"8 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599268","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}
Pub Date : 2024-04-06DOI: 10.1142/s1793042124500593
Elif Kızıldere Mutlu, Gökhan Soydan
<p>Denote by <span><math altimg="eq-00001.gif" display="inline" overflow="scroll"><mi>h</mi><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mo stretchy="false">−</mo><mi>p</mi><mo stretchy="false">)</mo></math></span><span></span> the class number of the imaginary quadratic field <span><math altimg="eq-00002.gif" display="inline" overflow="scroll"><mi>ℚ</mi><mo stretchy="false">(</mo><msqrt><mrow><mo stretchy="false">−</mo><mi>p</mi></mrow></msqrt><mo stretchy="false">)</mo></math></span><span></span> with <span><math altimg="eq-00003.gif" display="inline" overflow="scroll"><mi>p</mi></math></span><span></span> prime. It is well known that <span><math altimg="eq-00004.gif" display="inline" overflow="scroll"><mi>h</mi><mo>=</mo><mn>1</mn></math></span><span></span> for <span><math altimg="eq-00005.gif" display="inline" overflow="scroll"><mi>p</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>1</mn><mn>1</mn><mo>,</mo><mn>1</mn><mn>9</mn><mo>,</mo><mn>4</mn><mn>3</mn><mo>,</mo><mn>6</mn><mn>7</mn><mo>,</mo><mn>1</mn><mn>6</mn><mn>3</mn><mo stretchy="false">}</mo></math></span><span></span>. Recently, all the solutions of the Diophantine equation <span><math altimg="eq-00006.gif" display="inline" overflow="scroll"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>=</mo><mn>4</mn><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> with <span><math altimg="eq-00007.gif" display="inline" overflow="scroll"><mi>h</mi><mo>=</mo><mn>1</mn></math></span><span></span> were given by Chakraborty <i>et al</i>. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, <i>Publ. Math. Debrecen</i><b>97</b>(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation <span><math altimg="eq-00008.gif" display="inline" overflow="scroll"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> in unknown integers <span><math altimg="eq-00009.gif" display="inline" overflow="scroll"><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo></math></span><span></span> where <span><math altimg="eq-00010.gif" display="inline" overflow="scroll"><mi>s</mi><mo>≥</mo><mn>0</mn></math></span><span></span>, <span><math altimg="eq-00011.gif" display="inline" overflow="scroll"><mi>r</mi><mo>≥</mo><mn>3</mn></math></span><span></span>, <span><math altimg="eq-00012.gif" display="inline" overflow="scroll"><mi>n</mi><mo>≥</mo><mn>3</mn></math></span><span></span>, <span><math altimg="eq-00013.gif" display="inline" overflow="scroll"><mi>h</mi><mo>∈</mo><mo st
用 h=h(-p) 表示 p 为素数的虚二次型域ℚ(-p) 的类数。众所周知，对于 p∈{3,7,11,19,43,67,163}，h=1。最近，Chakraborty 等人在 [Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations, Publ.Math.Debrecen97(3-4) (2020) 339-352] 中给出。本文研究未知整数 (x,y,s,r,n) 中的二叉方程 x2+ps=2ryn，其中 s≥0，r≥3，n≥3，h∈{1,2,3} 和 gcd(x,y)=1。为此，我们使用了与弗雷-赫勒高椭圆曲线相关的伽罗瓦表示的模块性、交映方法和经典代数数论的基本方法的已知结果。本文的目的是扩展 Chakraborty 等人的上述结果。
{"title":"On the solutions of some Lebesgue–Ramanujan–Nagell type equations","authors":"Elif Kızıldere Mutlu, Gökhan Soydan","doi":"10.1142/s1793042124500593","DOIUrl":"https://doi.org/10.1142/s1793042124500593","url":null,"abstract":"<p>Denote by <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>h</mi><mo>=</mo><mi>h</mi><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">−</mo><mi>p</mi><mo stretchy=\"false\">)</mo></math></span><span></span> the class number of the imaginary quadratic field <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>ℚ</mi><mo stretchy=\"false\">(</mo><msqrt><mrow><mo stretchy=\"false\">−</mo><mi>p</mi></mrow></msqrt><mo stretchy=\"false\">)</mo></math></span><span></span> with <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span> prime. It is well known that <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>h</mi><mo>=</mo><mn>1</mn></math></span><span></span> for <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi><mo>∈</mo><mo stretchy=\"false\">{</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>1</mn><mn>1</mn><mo>,</mo><mn>1</mn><mn>9</mn><mo>,</mo><mn>4</mn><mn>3</mn><mo>,</mo><mn>6</mn><mn>7</mn><mo>,</mo><mn>1</mn><mn>6</mn><mn>3</mn><mo stretchy=\"false\">}</mo></math></span><span></span>. Recently, all the solutions of the Diophantine equation <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>=</mo><mn>4</mn><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> with <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>h</mi><mo>=</mo><mn>1</mn></math></span><span></span> were given by Chakraborty <i>et al</i>. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, <i>Publ. Math. Debrecen</i><b>97</b>(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> in unknown integers <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>n</mi><mo stretchy=\"false\">)</mo><mo>,</mo></math></span><span></span> where <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>s</mi><mo>≥</mo><mn>0</mn></math></span><span></span>, <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>r</mi><mo>≥</mo><mn>3</mn></math></span><span></span>, <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≥</mo><mn>3</mn></math></span><span></span>, <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mi>h</mi><mo>∈</mo><mo st","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"30 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599270","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}
Pub Date : 2024-04-06DOI: 10.1142/s1793042124500805
Jun-Hwi Min
We prove the best possible upper bounds of the gaps between non-vanishing Fourier coefficients of half-integral weight cuspforms. This improves the works of Balog–Ono and Thorner. We also show an asymptotic formula of central modular -values for short intervals.
我们证明了半整数权凹凸形非相等傅里叶系数之间间隙的最佳上限。这改进了 Balog-Ono 和 Thorner 的工作。我们还展示了短区间中心模态 L 值的渐近公式。
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Pub Date : 2024-04-06DOI: 10.1142/s1793042124500799
Subham Bhakta, Srilakshmi Krishnamoorthy, R. Muneeswaran
Serre showed that for any integer for almost all where is the Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study over the set of integers with many prime factors. Moreover, we show that any residue class can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of
塞雷证明，对于任意整数 m，几乎所有 n 的 a(n)≡0(modm)，其中 a(n) 是任意有理系数模形式的第 n 个傅里叶系数。在本文中，我们考虑了某类余弦形式，并研究了在具有 O(1) 多质因数的整数集合上 #{a(n)(modm)}n≤x 的问题。此外，我们还证明了任何残差类 a∈ℤ/mℤ 都可以写成最多 13 个傅里叶系数之和，而这些系数作为 m 的函数是多项式有界的。
{"title":"Congruence classes for modular forms over small sets","authors":"Subham Bhakta, Srilakshmi Krishnamoorthy, R. Muneeswaran","doi":"10.1142/s1793042124500799","DOIUrl":"https://doi.org/10.1142/s1793042124500799","url":null,"abstract":"<p>Serre showed that for any integer <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>m</mi><mo>,</mo><mspace width=\"0.25em\"></mspace><mi>a</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo><mo>≡</mo><mn>0</mn><mspace width=\"0.3em\"></mspace><mo stretchy=\"false\">(</mo><mo>mod</mo><mspace width=\"0.3em\"></mspace><mi>m</mi><mo stretchy=\"false\">)</mo></math></span><span></span> for almost all <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>,</mo></math></span><span></span> where <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>a</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is the <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mstyle><mtext>th</mtext></mstyle></math></span><span></span> Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>#</mi><msub><mrow><mo stretchy=\"false\">{</mo><mi>a</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo><mspace width=\"0.3em\"></mspace><mo stretchy=\"false\">(</mo><mo>mod</mo><mspace width=\"0.3em\"></mspace><mi>m</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">}</mo></mrow><mrow><mi>n</mi><mo>≤</mo><mi>x</mi></mrow></msub></math></span><span></span> over the set of integers with <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>O</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math></span><span></span> many prime factors. Moreover, we show that any residue class <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>a</mi><mo>∈</mo><mi>ℤ</mi><mo stretchy=\"false\">/</mo><mi>m</mi><mi>ℤ</mi></math></span><span></span> can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>m</mi><mo>.</mo></math></span><span></span></p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"48 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599462","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}
Pub Date : 2024-04-06DOI: 10.1142/s1793042124500738
Peng Gao, Liangyi Zhao
In this paper, we establish the expected order of magnitude of the th-moment of central values of the family of Dirichlet -functions to a fixed prime modulus over function fields for all real .
在本文中，我们建立了在所有实数 k≥0 的函数域上，对固定素模的 Dirichlet L 函数族中心值的第 k 次矩阵的预期数量级。
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