Nicolas Berkopec, Jacob Branch, Rachel Heikkinen, C. Nunn, T. Wong
Elliptic Dedekind sums were introduced by R. Sczech as generalizations of classical Dedekind sums to complex lattices. We show that for any lattice with real $j$-invariant, the values of suitably normalized elliptic Dedekind sums are dense in the real numbers. This extends an earlier result of Ito for Euclidean imaginary quadratic rings. Our proof is an adaptation of the recent work of Kohnen, which gives a new proof of the density of values of classical Dedekind sums.
{"title":"The density of elliptic Dedekind sums","authors":"Nicolas Berkopec, Jacob Branch, Rachel Heikkinen, C. Nunn, T. Wong","doi":"10.4064/aa210921-27-7","DOIUrl":"https://doi.org/10.4064/aa210921-27-7","url":null,"abstract":"Elliptic Dedekind sums were introduced by R. Sczech as generalizations of classical Dedekind sums to complex lattices. We show that for any lattice with real $j$-invariant, the values of suitably normalized elliptic Dedekind sums are dense in the real numbers. This extends an earlier result of Ito for Euclidean imaginary quadratic rings. Our proof is an adaptation of the recent work of Kohnen, which gives a new proof of the density of values of classical Dedekind sums.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44111491","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 assuming RH that phenomena concerning pairs of zeros established $via$ pair correlations occur with positive density (with at most a slight adjustment of the constants). Also, while a double zero is commonly considered to be a close pair, we consider the difference between two $distinct$ zeros.
{"title":"Small gaps and small spacings between zeta zeros","authors":"H. Bui, D. Goldston, M. Milinovich, H. Montgomery","doi":"10.4064/aa220731-15-2","DOIUrl":"https://doi.org/10.4064/aa220731-15-2","url":null,"abstract":"We show assuming RH that phenomena concerning pairs of zeros established $via$ pair correlations occur with positive density (with at most a slight adjustment of the constants). Also, while a double zero is commonly considered to be a close pair, we consider the difference between two $distinct$ zeros.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46638781","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}
This paper addresses the factorization of polynomials of the form $F(x) = f_{0}(x) + f_{1}(x) x^{n} + cdots + f_{r-1}(x) x^{(r-1)n} + f_{r}(x) x^{rn}$ where $r$ is a fixed positive integer and the $f_{j}(x)$ are fixed polynomials in $mathbb Z[x]$ for $0 le j le r$. We provide an efficient method for showing that for $n$ sufficiently large and reasonable conditions on the $f_{j}(x)$, the non-reciprocal part of $F(x)$ is either $1$ or irreducible. We illustrate the approach including giving two examples that arise from trace fields of hyperbolic $3$-manifolds.
{"title":"On the factorization of lacunary polynomials","authors":"M. Filaseta","doi":"10.4064/aa220723-16-5","DOIUrl":"https://doi.org/10.4064/aa220723-16-5","url":null,"abstract":"This paper addresses the factorization of polynomials of the form $F(x) = f_{0}(x) + f_{1}(x) x^{n} + cdots + f_{r-1}(x) x^{(r-1)n} + f_{r}(x) x^{rn}$ where $r$ is a fixed positive integer and the $f_{j}(x)$ are fixed polynomials in $mathbb Z[x]$ for $0 le j le r$. We provide an efficient method for showing that for $n$ sufficiently large and reasonable conditions on the $f_{j}(x)$, the non-reciprocal part of $F(x)$ is either $1$ or irreducible. We illustrate the approach including giving two examples that arise from trace fields of hyperbolic $3$-manifolds.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49315281","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 determine all modular curves X +0 ( N ) that admit infinitely many cubic points over the rational field Q .
我们确定了在有理域Q上允许无限多个三次点的所有模曲线X+0（N）。
{"title":"Infinitely many cubic points for $X_0^+(N)$ over $mathbb Q$","authors":"Francesc Bars, Tarun Dalal","doi":"10.4064/aa220714-10-11","DOIUrl":"https://doi.org/10.4064/aa220714-10-11","url":null,"abstract":"We determine all modular curves X +0 ( N ) that admit infinitely many cubic points over the rational field Q .","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42078338","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 L be a primitive Gaussian line, that is, a line in the complex plane that contains two, and hence infinitely many, coprime Gaussian integers. We prove that there exists an integer G L such that for every integer n ≥ G L there are infinitely many sequences of n consecutive Gaussian integers on L with the property that none of the Gaussian integers in the sequence is coprime to all the others. We also investigate the smallest integer g L such that L contains a sequence of g L consecutive Gaussian integers with this property. We show that g L 6 = G L in general. Also, g L ≥ 7 for every Gaussian line L , and we give necessary and sufficient conditions for g L = 7 and describe infinitely many Gaussian lines with g L ≥ 260 , 000. We conjecture that both g L and G L can be arbitrarily large. Our results extend a well-known problem of Pillai from the rational integers to the Gaussian integers.
{"title":"Extending a problem of Pillai to Gaussian lines","authors":"E. Magness, Brian Nugent, L. Robertson","doi":"10.4064/aa220227-11-10","DOIUrl":"https://doi.org/10.4064/aa220227-11-10","url":null,"abstract":"Let L be a primitive Gaussian line, that is, a line in the complex plane that contains two, and hence infinitely many, coprime Gaussian integers. We prove that there exists an integer G L such that for every integer n ≥ G L there are infinitely many sequences of n consecutive Gaussian integers on L with the property that none of the Gaussian integers in the sequence is coprime to all the others. We also investigate the smallest integer g L such that L contains a sequence of g L consecutive Gaussian integers with this property. We show that g L 6 = G L in general. Also, g L ≥ 7 for every Gaussian line L , and we give necessary and sufficient conditions for g L = 7 and describe infinitely many Gaussian lines with g L ≥ 260 , 000. We conjecture that both g L and G L can be arbitrarily large. Our results extend a well-known problem of Pillai from the rational integers to the Gaussian integers.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46714648","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}
Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials $h$ for which there is an irreducible monic quadratic $g$ such that the product $gh$ is exceptional. We construct exceptional polynomials with all factors of the form $X^{p}-b$, $p$ prime and $b$ square free.
{"title":"On polynomials with roots modulo almost all primes","authors":"C. Elsholtz, Benjamin Klahn, Marc Technau","doi":"10.4064/aa220407-9-7","DOIUrl":"https://doi.org/10.4064/aa220407-9-7","url":null,"abstract":"Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials $h$ for which there is an irreducible monic quadratic $g$ such that the product $gh$ is exceptional. We construct exceptional polynomials with all factors of the form $X^{p}-b$, $p$ prime and $b$ square free.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42073882","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 consider some families of binary binomial forms $aX^d+bY^d$, with $a$ and $b$ integers. Under suitable assumptions, we prove that every rational integer $m$ with $|m|ge 2$ is only represented by a finite number of the forms of this family (with varying $d,a,b$). Furthermore {the number of such forms of degree $ge d_0$ representing $m$ is bounded by $O(|m|^{(1/d_0)+epsilon})$} uniformly for $vert m vert geq 2$. We also prove that the integers in the interval $[-N,N]$ represented by one of the form of the family with degree $dgeq d_0$ are almost all represented by some form of the family with degree $d=d_0$. In a previous {paper} we investigated the particular case where the binary binomial forms are positive definite. We now treat the general case by using a lower bound for linear forms of logarithms.
{"title":"Number of integers represented by\u0000families of binary forms (I)","authors":"'Etienne Fouvry, M. Waldschmidt","doi":"10.4064/aa220606-16-2","DOIUrl":"https://doi.org/10.4064/aa220606-16-2","url":null,"abstract":"We consider some families of binary binomial forms $aX^d+bY^d$, with $a$ and $b$ integers. Under suitable assumptions, we prove that every rational integer $m$ with $|m|ge 2$ is only represented by a finite number of the forms of this family (with varying $d,a,b$). Furthermore {the number of such forms of degree $ge d_0$ representing $m$ is bounded by $O(|m|^{(1/d_0)+epsilon})$} uniformly for $vert m vert geq 2$. We also prove that the integers in the interval $[-N,N]$ represented by one of the form of the family with degree $dgeq d_0$ are almost all represented by some form of the family with degree $d=d_0$. In a previous {paper} we investigated the particular case where the binary binomial forms are positive definite. We now treat the general case by using a lower bound for linear forms of logarithms.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46063301","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 study certain aspects of the Selberg sieve, in particular when sifting by rather thin sets of primes. We derive new results for the lower bound sieve suited especially for this setup and we apply them in particular to give a new sieve-propelled proof of Linnik’s theorem on the least prime in an arithmetic progression in the case of the presence of exceptional zeros.
{"title":"Selberg’s sieve of irregular density","authors":"J. Friedlander, H. Iwaniec","doi":"10.4064/aa220719-5-10","DOIUrl":"https://doi.org/10.4064/aa220719-5-10","url":null,"abstract":": We study certain aspects of the Selberg sieve, in particular when sifting by rather thin sets of primes. We derive new results for the lower bound sieve suited especially for this setup and we apply them in particular to give a new sieve-propelled proof of Linnik’s theorem on the least prime in an arithmetic progression in the case of the presence of exceptional zeros.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44594227","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 paper, we investigate the one-level density of low-lying zeros of quadratic twists of automorphic $L$-functions under the generalized Riemann hypothesis and the Ramanujan-Petersson conjecture. We improve upon the known results using only functional equations for quadratic Dirichlet $L$-functions.
{"title":"One-level density of quadratic twists of $L$-functions","authors":"Peng Gao, Liangyi Zhao","doi":"10.4064/aa220613-13-12","DOIUrl":"https://doi.org/10.4064/aa220613-13-12","url":null,"abstract":"In this paper, we investigate the one-level density of low-lying zeros of quadratic twists of automorphic $L$-functions under the generalized Riemann hypothesis and the Ramanujan-Petersson conjecture. We improve upon the known results using only functional equations for quadratic Dirichlet $L$-functions.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48585965","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}
This study investigates the existence of tuples $(k, ell, m)$ of integers such that all of $k$, $ell$, $m$, $k+ell$, $ell+m$, $m+k$, $k+ell+m$ belong to $S(alpha)$, where $S(alpha)$ is the set of all integers of the form $lfloor alpha n^2 rfloor$ for $ngeq alpha^{-1/2}$ and $lfloor xrfloor$ denotes the integer part of $x$. We show that $T(alpha)$, the set of all such tuples, is infinite for all $alphain (0,1)cap mathbb{Q}$ and for almost all $alphain (0,1)$ in the sense of the Lebesgue measure. Furthermore, we show that if there exists $alpha>0$ such that $T(alpha)$ is finite, then there is no perfect Euler brick. We also examine the set of all integers of the form $lceil alpha n^2 rceil$ for $nin mathbb{N}$.
{"title":"A system of certain linear Diophantine equations\u0000on analogs of squares","authors":"Yuya Kanado, Kota Saito","doi":"10.4064/aa220622-19-1","DOIUrl":"https://doi.org/10.4064/aa220622-19-1","url":null,"abstract":"This study investigates the existence of tuples $(k, ell, m)$ of integers such that all of $k$, $ell$, $m$, $k+ell$, $ell+m$, $m+k$, $k+ell+m$ belong to $S(alpha)$, where $S(alpha)$ is the set of all integers of the form $lfloor alpha n^2 rfloor$ for $ngeq alpha^{-1/2}$ and $lfloor xrfloor$ denotes the integer part of $x$. We show that $T(alpha)$, the set of all such tuples, is infinite for all $alphain (0,1)cap mathbb{Q}$ and for almost all $alphain (0,1)$ in the sense of the Lebesgue measure. Furthermore, we show that if there exists $alpha>0$ such that $T(alpha)$ is finite, then there is no perfect Euler brick. We also examine the set of all integers of the form $lceil alpha n^2 rceil$ for $nin mathbb{N}$.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46519444","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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