{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
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