Pub Date : 2024-03-26DOI: 10.1142/s1793042124500763
Stephan Baier, Arkaprava Bhandari
Keating and Rudnick [The variance of the number of prime polynomials in short intervals and in residue classes, Int. Math. Res. Not.2014(1) (2014) 259–288] derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus is a polynomial in such that . The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition.
Keating and Rudnick [The variance of the number of prime polynomials in short intervals and in residue classes, Int.Math.Res. Not.2014(1) (2014) 259-288]导出了函数场设置中算术级数和短区间中素数方差的渐近公式。在此,我们考虑计算算术级数和短区间交集中素数方差的混合问题。Keating 和 Rudnick 使用内卷将短区间转化为算术级数。我们沿用了他们的方法,但在算术级数中也应用了这种反卷。当模数 Q 是𝔽q[T]中的多项式时,Q(0)≠0,这样就产生了对偶算术级数。后者是我们在本文中始终保留的限制条件。最后,我们将讨论如何放宽这一条件。
{"title":"Variance of primes in short residue classes for function fields","authors":"Stephan Baier, Arkaprava Bhandari","doi":"10.1142/s1793042124500763","DOIUrl":"https://doi.org/10.1142/s1793042124500763","url":null,"abstract":"<p>Keating and Rudnick [The variance of the number of prime polynomials in short intervals and in residue classes, <i>Int. Math. Res. Not.</i><b>2014</b>(1) (2014) 259–288] derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>Q</mi></math></span><span></span> is a polynomial in <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">[</mo><mi>T</mi><mo stretchy=\"false\">]</mo></math></span><span></span> such that <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>Q</mi><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo><mo>≠</mo><mn>0</mn></math></span><span></span>. The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"53 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325446","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 : 2024-03-26DOI: 10.1142/s1793042124500684
Eberhard Freitag, Adrian Hauffe-Waschbüsch
Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus must be integral or half integral. Actually he proved that for a system of complex numbers of absolute value 1
can be an automorphy factor only if is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann.159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for and , Publ. Math. Inst. Hautes Études Sci.33 (1967) 59–137] of Bass–Milnor–Serre.
Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad.Sci. Paris287 (1978) 203-208] 中证明的。(另见 7.1 [R.Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds.H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp.实际上,他证明了对于绝对值复数系统 v(M) 1v(M)det(CZ+D)r(r∈ℝ)(0.1) 只有当 2r 是积分时才能成为自形因子。我们给出了一个不同的证明。它使用了门尼克的结果[Zur Theorie der Siegelschen Modulgruppe, Math. Ann.159 (1965) 115-129],即西格尔模群的有限指数子群是全等子群,以及[Solution of the congruence subgroup problem for SLn(n≥3) and Sp2n(n≥2), Publ.Math.高等科学研究所,33 (1967) 59-137]的 Bass-Milnor-Serre.
{"title":"Multiplier systems for Siegel modular groups","authors":"Eberhard Freitag, Adrian Hauffe-Waschbüsch","doi":"10.1142/s1793042124500684","DOIUrl":"https://doi.org/10.1142/s1793042124500684","url":null,"abstract":"<p>Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, <i>C. R. Acad. Sci. Paris</i><b>287</b> (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in <i>Automorphic Forms and Representations of Algebraic Groups Over Local Fields</i>, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>g</mi><mo>></mo><mn>1</mn></math></span><span></span> must be integral or half integral. Actually he proved that for a system <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>v</mi><mo stretchy=\"false\">(</mo><mi>M</mi><mo stretchy=\"false\">)</mo></math></span><span></span> of complex numbers of absolute value 1</p><p><span><math altimg=\"eq-00003.gif\" display=\"block\" overflow=\"scroll\"><mtable columnalign=\"left\"><mtr><mtd columnalign=\"right\"><mspace width=\"8.5pc\"></mspace><mi>v</mi><mo stretchy=\"false\">(</mo><mi>M</mi><mo stretchy=\"false\">)</mo><mo>det</mo><msup><mrow><mo stretchy=\"false\">(</mo><mi>C</mi><mi>Z</mi><mo stretchy=\"false\">+</mo><mi>D</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi>r</mi></mrow></msup><mo stretchy=\"false\">(</mo><mi>r</mi><mo>∈</mo><mi>ℝ</mi><mo stretchy=\"false\">)</mo><mspace width=\"8.5pc\"></mspace><mo stretchy=\"false\">(</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mtd><mtd></mtd></mtr></mtable></math></span><span></span></p><p>can be an automorphy factor only if <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn><mi>r</mi></math></span><span></span> is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, <i>Math. Ann.</i><b>159</b> (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mspace width=\".275em\"></mspace><mo stretchy=\"false\">(</mo><mi>n</mi><mo>≥</mo><mn>3</mn><mo stretchy=\"false\">)</mo></math></span><span></span> and <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mspace width=\".275em\"></mspace><mo stretchy=\"false\">(</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math></span><span></span>, <i>Publ. Math. Inst. Hautes Études Sci.</i><b>33</b> (1967) 59–137] of Bass–Milnor–Serre.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"252 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325840","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 : 2024-03-26DOI: 10.1142/s1793042124500714
Andrew Y. Z. Wang, Zheng Xu
In this work, we establish two interesting partition identities involving the minimal odd excludant, which has attracted great attention in recent years. In particular, we find a strong refinement of Euler’s celebrated theorem that the number of partitions of an integer into odd parts equals the number of partitions of that integer into distinct parts.
{"title":"The minimal odd excludant and Euler’s partition theorem","authors":"Andrew Y. Z. Wang, Zheng Xu","doi":"10.1142/s1793042124500714","DOIUrl":"https://doi.org/10.1142/s1793042124500714","url":null,"abstract":"<p>In this work, we establish two interesting partition identities involving the minimal odd excludant, which has attracted great attention in recent years. In particular, we find a strong refinement of Euler’s celebrated theorem that the number of partitions of an integer into odd parts equals the number of partitions of that integer into distinct parts.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"32 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325447","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 : 2024-03-26DOI: 10.1142/s1793042124500726
Brad Isaacson
We define a three-character analogue of the generalized Dedekind–Rademacher sum introduced by Hall, Wilson, and Zagier and prove its reciprocity formula which contains all of the reciprocity formulas in the literature for generalized Dedekind–Rademacher sums attached (and not attached) to Dirichlet characters as special cases. Additionally, we prove related polynomial reciprocity formulas which contain all of the polynomial reciprocity formulas in the literature as special cases, such as those given by Carlitz, Beck & Kohl, and the present author.
{"title":"Reciprocity formulae for generalized Dedekind–Rademacher sums attached to three Dirichlet characters and related polynomial reciprocity formulae","authors":"Brad Isaacson","doi":"10.1142/s1793042124500726","DOIUrl":"https://doi.org/10.1142/s1793042124500726","url":null,"abstract":"<p>We define a three-character analogue of the generalized Dedekind–Rademacher sum introduced by Hall, Wilson, and Zagier and prove its reciprocity formula which contains all of the reciprocity formulas in the literature for generalized Dedekind–Rademacher sums attached (and not attached) to Dirichlet characters as special cases. Additionally, we prove related polynomial reciprocity formulas which contain all of the polynomial reciprocity formulas in the literature as special cases, such as those given by Carlitz, Beck & Kohl, and the present author.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"28 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325449","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 : 2024-03-26DOI: 10.1142/s1793042124500751
Bin Chen
Let denote the divisor function and be an admissible set. We prove that there are infinitely many for which the product is square-free and , where is asymptotic to . It improves a previous result of Ram Murty and Vatwani, replacing by . The main ingredients in our proof are the higher rank Selberg sieve and Irving–Wu–Xi estimate for the divisor function in arithmetic progressions to smooth moduli.
{"title":"On almost-prime k-tuples","authors":"Bin Chen","doi":"10.1142/s1793042124500751","DOIUrl":"https://doi.org/10.1142/s1793042124500751","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>τ</mi></math></span><span></span> denote the divisor function and <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"cal\">ℋ</mi><mo>=</mo><mo stretchy=\"false\">{</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">}</mo></math></span><span></span> be an admissible set. We prove that there are infinitely many <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span> for which the product <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">+</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> is square-free and <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mi>τ</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">+</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy=\"false\">)</mo><mo>≤</mo><mo stretchy=\"false\">⌊</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">⌋</mo></math></span><span></span>, where <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>ρ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span><span></span> is asymptotic to <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mfrac><mrow><mn>2</mn><mn>1</mn><mn>2</mn><mn>6</mn></mrow><mrow><mn>2</mn><mn>8</mn><mn>5</mn><mn>3</mn></mrow></mfrac><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span></span>. It improves a previous result of Ram Murty and Vatwani, replacing <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn><mo stretchy=\"false\">/</mo><mn>4</mn></math></span><span></span> by <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn><mn>1</mn><mn>2</mn><mn>6</mn><mo stretchy=\"false\">/</mo><mn>2</mn><mn>8</mn><mn>5</mn><mn>3</mn></math></span><span></span>. The main ingredients in our proof are the higher rank Selberg sieve and Irving–Wu–Xi estimate for the divisor function in arithmetic progressions to smooth moduli.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"42 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325724","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 : 2024-03-26DOI: 10.1142/s1793042124500635
Piotr Miska, Maciej Ulas
In this paper, we investigate the set of positive integer solutions of the title Diophantine equation. In particular, for a given we prove boundedness of the number of solutions, give precise upper bound on the common value of and together with the biggest value of the variable appearing in the solution. Moreover, we enumerate all solutions for and discuss the set of values of over elements of .
{"title":"On the Diophantine equation σ2(X¯n) = σn(X¯n)","authors":"Piotr Miska, Maciej Ulas","doi":"10.1142/s1793042124500635","DOIUrl":"https://doi.org/10.1142/s1793042124500635","url":null,"abstract":"<p>In this paper, we investigate the set <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math></span><span></span> of positive integer solutions of the title Diophantine equation. In particular, for a given <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span> we prove boundedness of the number of solutions, give precise upper bound on the common value of <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mover accent=\"false\"><mrow><mi>X</mi></mrow><mo accent=\"true\">¯</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> and <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mover accent=\"false\"><mrow><mi>X</mi></mrow><mo accent=\"true\">¯</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> together with the biggest value of the variable <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> appearing in the solution. Moreover, we enumerate all solutions for <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≤</mo><mn>1</mn><mn>6</mn></math></span><span></span> and discuss the set of values of <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">/</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo stretchy=\"false\">−</mo><mn>1</mn></mrow></msub></math></span><span></span> over elements of <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math></span><span></span>.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"101 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325497","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 : 2024-03-21DOI: 10.1142/s1793042124500568
Weili Yao
In this paper, we investigate the square of the normalized Fourier coefficients of the primitive cusp forms and its symmetric-lift at integers with a fixed number of distinct prime divisors, and present asymptotic formulas for them in short intervals.
在本文中,我们研究了在具有固定数目的不同素除数的整数上,原始尖顶形式 f 的归一化傅里叶系数的平方及其对称提升,并给出了它们在短区间内的渐近公式。
{"title":"Fourier coefficients of cusp forms on special sequences","authors":"Weili Yao","doi":"10.1142/s1793042124500568","DOIUrl":"https://doi.org/10.1142/s1793042124500568","url":null,"abstract":"<p>In this paper, we investigate the square of the normalized Fourier coefficients of the primitive cusp forms <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi></math></span><span></span> and its symmetric-lift at integers with a fixed number of distinct prime divisors, and present asymptotic formulas for them in short intervals.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"158 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204341","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 : 2024-03-21DOI: 10.1142/s1793042124500581
Deepesh Singhal, Yuxin Lin
<p>We fix a number field <span><math altimg="eq-00003.gif" display="inline" overflow="scroll"><mi>K</mi></math></span><span></span> and study statistical properties of the ring <span><math altimg="eq-00004.gif" display="inline" overflow="scroll"><msub><mrow><mi mathvariant="cal">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">[</mo><mi>γ</mi><mo stretchy="false">]</mo><mo stretchy="false">∩</mo><mi>K</mi></math></span><span></span> as <span><math altimg="eq-00005.gif" display="inline" overflow="scroll"><mi>γ</mi></math></span><span></span> varies over algebraic numbers of a fixed degree <span><math altimg="eq-00006.gif" display="inline" overflow="scroll"><mi>n</mi><mo>≥</mo><mn>2</mn></math></span><span></span>. Given <span><math altimg="eq-00007.gif" display="inline" overflow="scroll"><mi>k</mi><mo>≥</mo><mn>1</mn></math></span><span></span>, we explicitly compute the density of <span><math altimg="eq-00008.gif" display="inline" overflow="scroll"><mi>γ</mi></math></span><span></span> for which <span><math altimg="eq-00009.gif" display="inline" overflow="scroll"><msub><mrow><mi mathvariant="cal">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">[</mo><mi>γ</mi><mo stretchy="false">]</mo><mo stretchy="false">∩</mo><mi>K</mi><mo>=</mo><msub><mrow><mi mathvariant="cal">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mi>k</mi><mo stretchy="false">]</mo></math></span><span></span> and show that this does not depend on the number field <span><math altimg="eq-00010.gif" display="inline" overflow="scroll"><mi>K</mi></math></span><span></span>. In particular, we show that the density of <span><math altimg="eq-00011.gif" display="inline" overflow="scroll"><mi>γ</mi></math></span><span></span> for which <span><math altimg="eq-00012.gif" display="inline" overflow="scroll"><msub><mrow><mi mathvariant="cal">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">[</mo><mi>γ</mi><mo stretchy="false">]</mo><mo stretchy="false">∩</mo><mi>K</mi><mo>=</mo><msub><mrow><mi mathvariant="cal">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span><span></span> is <span><math altimg="eq-00013.gif" display="inline" overflow="scroll"><mfrac><mrow><mi>ζ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>ζ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mfrac></math></span><span></span>. In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, <i>Int. J. Number Theory</i> (2023), doi:10.1142/S1793042124500167], the authors define <span><math altimg="eq-00014.gif" display="inline" overflow="scroll"><mi>X</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>γ</mi><mo stretchy="false">)</mo></math></span><span></span> to be a certain finite subset of <span><math altimg="eq-00015.gif" display="inline" overflow="scroll"><mstyle><mtext>Spec</mtext></mst
我们固定一个数域 K,研究当 γ 在固定度 n≥2 的代数数上变化时,环 𝒪K[γ]∩K 的统计性质。给定 k≥1,我们明确计算了 γ 的密度,其中 𝒪K[γ]∩K=𝒪K[1/k],并证明它不依赖于数域 K。特别是,我们证明了 γ 的密度,其中 𝒪K[γ]∩K=𝒪K 是 ζ(n+1)ζ(n)。在最近的一篇论文 [Singhal and Lin, Primes in denominators of algebraic numbers, Int.J. Number Theory (2023), doi:10.1142/S1793042124500167] 中,作者定义 X(K,γ) 为 Spec(𝒪K) 的某个有限子集,并证明 X(K,γ) 决定了环𝒪K[γ]∩K。我们证明,如果𝔭1,𝔭2∈Spec(𝒪K)满足𝔭1∩≠𝔭2∩ℤ,那么事件𝔭1∈X(K,γ)和𝔭2∈X(K,γ)是独立的。当 t→∞ 时,我们研究|X(K,γ)|=t 时 γ 密度的渐近线。
{"title":"Density questions in rings of the form 𝒪K[γ] ∩ K","authors":"Deepesh Singhal, Yuxin Lin","doi":"10.1142/s1793042124500581","DOIUrl":"https://doi.org/10.1142/s1793042124500581","url":null,"abstract":"<p>We fix a number field <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>K</mi></math></span><span></span> and study statistical properties of the ring <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy=\"false\">[</mo><mi>γ</mi><mo stretchy=\"false\">]</mo><mo stretchy=\"false\">∩</mo><mi>K</mi></math></span><span></span> as <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>γ</mi></math></span><span></span> varies over algebraic numbers of a fixed degree <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≥</mo><mn>2</mn></math></span><span></span>. Given <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi><mo>≥</mo><mn>1</mn></math></span><span></span>, we explicitly compute the density of <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>γ</mi></math></span><span></span> for which <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy=\"false\">[</mo><mi>γ</mi><mo stretchy=\"false\">]</mo><mo stretchy=\"false\">∩</mo><mi>K</mi><mo>=</mo><msub><mrow><mi mathvariant=\"cal\">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy=\"false\">[</mo><mn>1</mn><mo stretchy=\"false\">/</mo><mi>k</mi><mo stretchy=\"false\">]</mo></math></span><span></span> and show that this does not depend on the number field <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>K</mi></math></span><span></span>. In particular, we show that the density of <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>γ</mi></math></span><span></span> for which <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi mathvariant=\"cal\">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy=\"false\">[</mo><mi>γ</mi><mo stretchy=\"false\">]</mo><mo stretchy=\"false\">∩</mo><mi>K</mi><mo>=</mo><msub><mrow><mi mathvariant=\"cal\">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span><span></span> is <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mfrac><mrow><mi>ζ</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">+</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow><mrow><mi>ζ</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></mrow></mfrac></math></span><span></span>. In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, <i>Int. J. Number Theory</i> (2023), doi:10.1142/S1793042124500167], the authors define <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><mi>X</mi><mo stretchy=\"false\">(</mo><mi>K</mi><mo>,</mo><mi>γ</mi><mo stretchy=\"false\">)</mo></math></span><span></span> to be a certain finite subset of <span><math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext>Spec</mtext></mst","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"15 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204336","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 : 2024-03-20DOI: 10.1142/s1793042124500659
Byungchan Kim, Hayan Nam, Myungjun Yu
We introduce the multi-Gaussian polynomial , a multi-partition analogue of the Gaussian polynomial (also known as -binomial coefficient), as the generating function for certain restricted multi-color partitions. We study basic properties of multi-Gaussian polynomials and non-symmetric properties of . We also derive a Sylvester-type identity and its application.
{"title":"Multi-partition analogue of q-binomial coefficients","authors":"Byungchan Kim, Hayan Nam, Myungjun Yu","doi":"10.1142/s1793042124500659","DOIUrl":"https://doi.org/10.1142/s1793042124500659","url":null,"abstract":"<p>We introduce the multi-Gaussian polynomial <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>M</mi><mo>,</mo><mi>N</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, a multi-partition analogue of the Gaussian polynomial (also known as <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>q</mi></math></span><span></span>-binomial coefficient), as the generating function for certain restricted multi-color partitions. We study basic properties of multi-Gaussian polynomials and non-symmetric properties of <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>M</mi><mo>,</mo><mi>N</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. We also derive a Sylvester-type identity and its application.</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":"140204254","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 : 2024-03-20DOI: 10.1142/s1793042124500489
Sudesh Kaur Khanduja
We point out that there is an error in the proof of Theorem 1.1 in [The discriminant of compositum of algebraic number fields, Int. J. Number Theory15 (2019) 353–360]. We also prove that the result of this theorem holds with an additional hypothesis. However, it is an open problem whether the result of the theorem is true in general or not.
我们指出[The discriminant of compositum of algebraic number fields, Int. J Number Theory15 (2019)] 中定理 1.1 的证明有误。J. Number Theory15 (2019) 353-360]中的定理 1.1 的证明有误。我们还证明了该定理的结果在附加假设的情况下成立。然而,该定理的结果在一般情况下是否成立还是一个悬而未决的问题。
{"title":"Corrigendum to “The discriminant of compositum of algebraic number fields”","authors":"Sudesh Kaur Khanduja","doi":"10.1142/s1793042124500489","DOIUrl":"https://doi.org/10.1142/s1793042124500489","url":null,"abstract":"<p>We point out that there is an error in the proof of Theorem 1.1 in [The discriminant of compositum of algebraic number fields, <i>Int. J. Number Theory</i><b>15</b> (2019) 353–360]. We also prove that the result of this theorem holds with an additional hypothesis. However, it is an open problem whether the result of the theorem is true in general or not.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"2016 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140204333","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}