Pub Date : 2023-10-13DOI: 10.1142/s1793042124500192
Bernhard Heim, Markus Neuhauser
We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let ${g_d(n)}_{dgeq 0,n geq 1}$ be the double sequences $sigma_d(n)= sum_{ell mid n} ell^d$ or $psi_d(n)= n^d$. We associate double sequences $left{ p^{g_{d} }left( nright) right}$ and $left{ q^{g_{d} }left( nright) right} $, defined as the coefficients of begin{eqnarray*} sum_{n=0}^{infty} p^{g_{d} }left( nright) , t^{n}&:=&prod_{n=1}^{infty} left( 1 - t^{n} right)^{-frac{ sum_{ell mid n} mu(ell) , g_d(n/ell) }{n} }, sum_{n=0}^{infty} q^{g_{d} }left( nright) , t^{n}&:=&frac{1}{1 - sum_{n=1}^{infty} g_d(n) , t^{n} }. end{eqnarray*} These coefficients are related to the number of partitions $mathrm{p}left( nright) = p^{sigma _{1 }}left ( nright) $, plane partitions $ppleft( nright) = p^{sigma _{2 }}left( nright) $ of $n$, and Fibonacci numbers $F_{2n} = q^{psi _{1 }}left( nright) $. Let $n geq 3$ and let $n equiv 0 pmod{3}$. Then the coefficients are log-concave at $n$ for almost all $d$ in the exponential and geometric cases. The coefficients are not log-concave for almost all $d$ in both cases, if $n equiv 2 pmod{3}$. Let $nequiv 1 pmod{3}$. Then the log-concave property flips for almost all $d$.
{"title":"Log-Concavity of Infinite Product and Infinite Sum Generating Functions","authors":"Bernhard Heim, Markus Neuhauser","doi":"10.1142/s1793042124500192","DOIUrl":"https://doi.org/10.1142/s1793042124500192","url":null,"abstract":"We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let ${g_d(n)}_{dgeq 0,n geq 1}$ be the double sequences $sigma_d(n)= sum_{ell mid n} ell^d$ or $psi_d(n)= n^d$. We associate double sequences $left{ p^{g_{d} }left( nright) right}$ and $left{ q^{g_{d} }left( nright) right} $, defined as the coefficients of begin{eqnarray*} sum_{n=0}^{infty} p^{g_{d} }left( nright) , t^{n}&:=&prod_{n=1}^{infty} left( 1 - t^{n} right)^{-frac{ sum_{ell mid n} mu(ell) , g_d(n/ell) }{n} }, sum_{n=0}^{infty} q^{g_{d} }left( nright) , t^{n}&:=&frac{1}{1 - sum_{n=1}^{infty} g_d(n) , t^{n} }. end{eqnarray*} These coefficients are related to the number of partitions $mathrm{p}left( nright) = p^{sigma _{1 }}left ( nright) $, plane partitions $ppleft( nright) = p^{sigma _{2 }}left( nright) $ of $n$, and Fibonacci numbers $F_{2n} = q^{psi _{1 }}left( nright) $. Let $n geq 3$ and let $n equiv 0 pmod{3}$. Then the coefficients are log-concave at $n$ for almost all $d$ in the exponential and geometric cases. The coefficients are not log-concave for almost all $d$ in both cases, if $n equiv 2 pmod{3}$. Let $nequiv 1 pmod{3}$. Then the log-concave property flips for almost all $d$.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918468","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 : 2023-10-13DOI: 10.1142/s1793042124500386
Byoung Du Kim
{"title":"Iwasawa theory of plus/minus Selmer groups with non-co-free plus/minus local conditions","authors":"Byoung Du Kim","doi":"10.1142/s1793042124500386","DOIUrl":"https://doi.org/10.1142/s1793042124500386","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918474","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 : 2023-10-13DOI: 10.1142/s1793042124500222
William Frendreiss, Jennifer Gao, Austin Lei, Amy Woodall, Hui Xue, Daozhou Zhu
{"title":"Interlacing Properties for Zeros of a Family of Modular Forms","authors":"William Frendreiss, Jennifer Gao, Austin Lei, Amy Woodall, Hui Xue, Daozhou Zhu","doi":"10.1142/s1793042124500222","DOIUrl":"https://doi.org/10.1142/s1793042124500222","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918482","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 : 2023-08-04DOI: 10.1142/s1793042124500088
Xuejun Guo, Q. Ji, Hang Liu, H. Qin
{"title":"The mahler measure of x + 1/x + y + 1/y + 4 ± 42 and Beilinson’s conjecture","authors":"Xuejun Guo, Q. Ji, Hang Liu, H. Qin","doi":"10.1142/s1793042124500088","DOIUrl":"https://doi.org/10.1142/s1793042124500088","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82963127","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 : 2023-08-04DOI: 10.1142/s179304212350121x
Julia Q. D. Du, Dazhao Tang
{"title":"Vanishing coefficients in quotients of theta functions with odd moduli","authors":"Julia Q. D. Du, Dazhao Tang","doi":"10.1142/s179304212350121x","DOIUrl":"https://doi.org/10.1142/s179304212350121x","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79761386","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 : 2023-08-04DOI: 10.1142/s1793042123501233
Henry H. Kim
{"title":"Fourier coefficients of automorphic L-functions","authors":"Henry H. Kim","doi":"10.1142/s1793042123501233","DOIUrl":"https://doi.org/10.1142/s1793042123501233","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73411188","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 : 2023-08-04DOI: 10.1142/s1793042123501245
A. Bandini, M. Valentino
{"title":"Fourier coefficients and slopes of drinfeld modular forms","authors":"A. Bandini, M. Valentino","doi":"10.1142/s1793042123501245","DOIUrl":"https://doi.org/10.1142/s1793042123501245","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86187942","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 : 2023-08-04DOI: 10.1142/s179304212450009x
Debargha Banerjee, Vivek Rai
{"title":"Towards A mod-p lubin-tate theory for GL2 over totally real fields","authors":"Debargha Banerjee, Vivek Rai","doi":"10.1142/s179304212450009x","DOIUrl":"https://doi.org/10.1142/s179304212450009x","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78263434","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 : 2023-08-04DOI: 10.1142/s1793042124500052
Chunlin Wang, Liping Yang
{"title":"Newton polygons of L-functions for two-variable generalized kloosterman sums","authors":"Chunlin Wang, Liping Yang","doi":"10.1142/s1793042124500052","DOIUrl":"https://doi.org/10.1142/s1793042124500052","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85115281","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 : 2023-08-04DOI: 10.1142/s1793042124500076
Yong-Xiong Li
{"title":"Some remarks on p-adic L-functions for certain abelian varieties with complex multiplication","authors":"Yong-Xiong Li","doi":"10.1142/s1793042124500076","DOIUrl":"https://doi.org/10.1142/s1793042124500076","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78627505","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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