{"title":"Powers from products of terms in progressions with gaps","authors":"Michael A. Bennett","doi":"10.4064/aa220811-13-9","DOIUrl":"https://doi.org/10.4064/aa220811-13-9","url":null,"abstract":"","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134884236","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}
For a positive integer $tgeq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a non-negative integer $n$. In a recent paper, Keith and Zanello (2022) investigated the parity of $b_{t}(n)$ when $tleq 28$. They discovered new infinite fam
{"title":"Certain Diophantine equations and new parity results for 21-regular partitions","authors":"Ajit Singh, Gurinder Singh, Rupam Barman","doi":"10.4064/aa230203-5-7","DOIUrl":"https://doi.org/10.4064/aa230203-5-7","url":null,"abstract":"For a positive integer $tgeq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a non-negative integer $n$. In a recent paper, Keith and Zanello (2022) investigated the parity of $b_{t}(n)$ when $tleq 28$. They discovered new infinite fam","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"242 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135440483","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}
For $d gt 1$ a cubefree rational integer, we define an $L$-function (denoted $L_d(s)$) whose coefficients are derived from the cubic theta function for $mathbb Q(sqrt {-3})$. The Dirichlet series defining $L_d(s)$ converges for ${rm Re}(s) gt 1$, and
{"title":"The cubic Pell equation $L$-function","authors":"Dorian Goldfeld, Gerhardt Hinkle","doi":"10.4064/aa220918-18-8","DOIUrl":"https://doi.org/10.4064/aa220918-18-8","url":null,"abstract":"For $d gt 1$ a cubefree rational integer, we define an $L$-function (denoted $L_d(s)$) whose coefficients are derived from the cubic theta function for $mathbb Q(sqrt {-3})$. The Dirichlet series defining $L_d(s)$ converges for ${rm Re}(s) gt 1$, and","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135839483","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}
In this article, we extend our recent work on a Bombieri–Vinogradov-type theorem for sparse sets of prime powers $p^Nleqslant x^{1/4-varepsilon }$ with $pleqslant (log x)^C$ to sparse sets of moduli $sleqslant x^{1/3-varepsilon }$ with radical rad$(
{"title":"A Bombieri–Vinogradov-type theorem for moduli with small radical","authors":"Stephan Baier, Sudhir Pujahari","doi":"10.4064/aa221211-1-9","DOIUrl":"https://doi.org/10.4064/aa221211-1-9","url":null,"abstract":"In this article, we extend our recent work on a Bombieri–Vinogradov-type theorem for sparse sets of prime powers $p^Nleqslant x^{1/4-varepsilon }$ with $pleqslant (log x)^C$ to sparse sets of moduli $sleqslant x^{1/3-varepsilon }$ with radical rad$(","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135263345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"$varOmega$-result for the remainder term in Beurling’s prime number theorem for well-behaved integers","authors":"T. Hilberdink, Laima Kaziulytė","doi":"10.4064/aa220516-20-3","DOIUrl":"https://doi.org/10.4064/aa220516-20-3","url":null,"abstract":"","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70440102","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}
Let $mathcal O$ be an order in an algebraic number field and suppose that the set of distances $varDelta (mathcal O)$ of $mathcal O$ is nonempty (equivalently, $mathcal O$ is not half-factorial). If $mathcal O$ is seminormal (in particular, if $mat
{"title":"On orders in quadratic number fields with unusual sets of distances","authors":"Andreas Reinhart","doi":"10.4064/aa230515-4-10","DOIUrl":"https://doi.org/10.4064/aa230515-4-10","url":null,"abstract":"Let $mathcal O$ be an order in an algebraic number field and suppose that the set of distances $varDelta (mathcal O)$ of $mathcal O$ is nonempty (equivalently, $mathcal O$ is not half-factorial). If $mathcal O$ is seminormal (in particular, if $mat","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135261686","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 “Explicit estimates for the summatory function of ${{varLambda }}(n)/n$ from the one of ${{varLambda }}(n)$” (Acta Arith. 159 (2013), 113–122)","authors":"Olivier Ramaré","doi":"10.4064/aa220605-11-10","DOIUrl":"https://doi.org/10.4064/aa220605-11-10","url":null,"abstract":"","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135318029","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}
We show that, under some mild conditions, the orbit of an algebraic number under random iterations cannot approach another algebraic number very fast. As an application of this result, we prove that, in certain cases, all but finitely many terms in such a
{"title":"Diophantine approximation and primitive prime divisors in random iterations","authors":"Ruofan Li","doi":"10.4064/aa230303-12-8","DOIUrl":"https://doi.org/10.4064/aa230303-12-8","url":null,"abstract":"We show that, under some mild conditions, the orbit of an algebraic number under random iterations cannot approach another algebraic number very fast. As an application of this result, we prove that, in certain cases, all but finitely many terms in such a","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135704395","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}
Let $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$. This problem is connected with deriving an upper bound on the Petersson no
{"title":"Quaternary quadratic forms with prime discriminant","authors":"Jeremy Rouse, Katherine Thompson","doi":"10.4064/aa220601-14-7","DOIUrl":"https://doi.org/10.4064/aa220601-14-7","url":null,"abstract":"Let $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$. This problem is connected with deriving an upper bound on the Petersson no","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009415","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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