Pub Date : 2022-11-28DOI: 10.1017/S0305004122000457
A. Sahay
Abstract We study the moments $M_k(T;,alpha) = int_T^{2T} |zeta(s,alpha)|^{2k},dt$ of the Hurwitz zeta function $zeta(s,alpha)$ on the critical line, $s = 1/2 + it$ with a rational shift $alpha in mathbb{Q}$ . We conjecture, in analogy with the Riemann zeta function, that $M_k(T;,alpha) sim c_k(alpha) T (!log T)^{k^2}$ . Using heuristics from analytic number theory and random matrix theory, we conjecturally compute $c_k(alpha)$ . In the process, we investigate moments of products of Dirichlet L-functions on the critical line. We prove some of our conjectures for the cases $k = 1,2$ .
摘要研究了Hurwitz zeta函数$zeta(s,alpha)$在临界线上的矩$M_k(T;,alpha) = int_T^{2T} |zeta(s,alpha)|^{2k},dt$, $s = 1/2 + it$有一个合理的位移$alpha in mathbb{Q}$。我们推测，与黎曼函数类似，$M_k(T;,alpha) sim c_k(alpha) T (!log T)^{k^2}$。利用解析数论和随机矩阵理论的启发式方法，我们推测计算$c_k(alpha)$。在此过程中，我们研究了狄利克雷l函数在临界线上积的矩。我们对这些案例证明了我们的一些猜想$k = 1,2$。
{"title":"Moments of the Hurwitz zeta function on the critical line","authors":"A. Sahay","doi":"10.1017/S0305004122000457","DOIUrl":"https://doi.org/10.1017/S0305004122000457","url":null,"abstract":"Abstract We study the moments \u0000$M_k(T;,alpha) = int_T^{2T} |zeta(s,alpha)|^{2k},dt$\u0000 of the Hurwitz zeta function \u0000$zeta(s,alpha)$\u0000 on the critical line, \u0000$s = 1/2 + it$\u0000 with a rational shift \u0000$alpha in mathbb{Q}$\u0000 . We conjecture, in analogy with the Riemann zeta function, that \u0000$M_k(T;,alpha) sim c_k(alpha) T (!log T)^{k^2}$\u0000 . Using heuristics from analytic number theory and random matrix theory, we conjecturally compute \u0000$c_k(alpha)$\u0000 . In the process, we investigate moments of products of Dirichlet L-functions on the critical line. We prove some of our conjectures for the cases \u0000$k = 1,2$\u0000 .","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"1 1","pages":"631 - 661"},"PeriodicalIF":0.8,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76782575","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 : 2022-11-28DOI: 10.1017/s0305004122000469
ANDREA AVENI, PAOLO LEONETTI
We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers $x in (0,1]$ with the following property is comeager: for all integers $bge 2$ and $kge 1$ , the sequence of vectors made by the frequencies of all possibile strings of length k in the b-adic representation of x has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen (2004) in this Journal. We provide analogues in the context of analytic P-ideals and regular matrices.
{"title":"Most numbers are not normal","authors":"ANDREA AVENI, PAOLO LEONETTI","doi":"10.1017/s0305004122000469","DOIUrl":"https://doi.org/10.1017/s0305004122000469","url":null,"abstract":"<p>We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230612115128646-0745:S0305004122000469:S0305004122000469_inline1.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$x in (0,1]$\u0000</span></span>\u0000</span>\u0000</span> with the following property is comeager: for all integers <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230612115128646-0745:S0305004122000469:S0305004122000469_inline2.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$bge 2$\u0000</span></span>\u0000</span>\u0000</span> and <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230612115128646-0745:S0305004122000469:S0305004122000469_inline3.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$kge 1$\u0000</span></span>\u0000</span>\u0000</span>, the sequence of vectors made by the frequencies of all possibile strings of length <span>k</span> in the <span>b</span>-adic representation of <span>x</span> has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen (2004) in this Journal. We provide analogues in the context of analytic P-ideals and regular matrices.</p>","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"69 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138507948","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 : 2022-11-14DOI: 10.1017/s0305004123000415
Alina Ostafe, I. Shparlinski, J. Voloch
We obtain new bounds on short Weil sums over small multiplicative subgroups of prime finite fields which remain nontrivial in the range the classical Weil bound is already trivial. The method we use is a blend of techniques coming from algebraic geometry and additive combinatorics.
{"title":"Weil Sums over Small Subgroups","authors":"Alina Ostafe, I. Shparlinski, J. Voloch","doi":"10.1017/s0305004123000415","DOIUrl":"https://doi.org/10.1017/s0305004123000415","url":null,"abstract":"\u0000 We obtain new bounds on short Weil sums over small multiplicative subgroups of prime finite fields which remain nontrivial in the range the classical Weil bound is already trivial. The method we use is a blend of techniques coming from algebraic geometry and additive combinatorics.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80243682","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 : 2022-10-21DOI: 10.1017/S0305004123000154
Be'eri Greenfeld
Abstract We prove that there exist finitely generated, stably finite algebras which are not linear sofic. This was left open by Arzhantseva and Păunescu in 2017.
{"title":"Stable finiteness does not imply linear soficity","authors":"Be'eri Greenfeld","doi":"10.1017/S0305004123000154","DOIUrl":"https://doi.org/10.1017/S0305004123000154","url":null,"abstract":"Abstract We prove that there exist finitely generated, stably finite algebras which are not linear sofic. This was left open by Arzhantseva and Păunescu in 2017.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"30 1","pages":"319 - 325"},"PeriodicalIF":0.8,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87351527","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 : 2022-10-19DOI: 10.1017/s0305004122000408
{"title":"PSP volume 173 issue 3 Cover and Back matter","authors":"","doi":"10.1017/s0305004122000408","DOIUrl":"https://doi.org/10.1017/s0305004122000408","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"37 1","pages":"b1 - b6"},"PeriodicalIF":0.8,"publicationDate":"2022-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85940940","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 : 2022-10-19DOI: 10.1017/s0305004122000391
{"title":"PSP volume 173 issue 3 Cover and Front matter","authors":"","doi":"10.1017/s0305004122000391","DOIUrl":"https://doi.org/10.1017/s0305004122000391","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"5 1","pages":"f1 - f2"},"PeriodicalIF":0.8,"publicationDate":"2022-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88118418","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 : 2022-10-05DOI: 10.1017/S0305004123000178
Bhishan Jacelon
Abstract What is the probability that a random UHF algebra is of infinite type? What is the probability that a random simple AI algebra has at most k extremal traces? What is the expected value of the radius of comparison of a random Villadsen-type AH algebra? What is the probability that such an algebra is $mathcal{Z}$ -stable? What is the probability that a random Cuntz–Krieger algebra is purely infinite and simple, and what can be said about the distribution of its K-theory? By constructing $mathrm{C}^*$ -algebras associated with suitable random (walks on) graphs, we provide context in which these are meaningful questions with computable answers.
{"title":"Random amenable C*-algebras","authors":"Bhishan Jacelon","doi":"10.1017/S0305004123000178","DOIUrl":"https://doi.org/10.1017/S0305004123000178","url":null,"abstract":"Abstract What is the probability that a random UHF algebra is of infinite type? What is the probability that a random simple AI algebra has at most k extremal traces? What is the expected value of the radius of comparison of a random Villadsen-type AH algebra? What is the probability that such an algebra is \u0000$mathcal{Z}$\u0000 -stable? What is the probability that a random Cuntz–Krieger algebra is purely infinite and simple, and what can be said about the distribution of its K-theory? By constructing \u0000$mathrm{C}^*$\u0000 -algebras associated with suitable random (walks on) graphs, we provide context in which these are meaningful questions with computable answers.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"30 1","pages":"345 - 366"},"PeriodicalIF":0.8,"publicationDate":"2022-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86746099","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 : 2022-08-17DOI: 10.1017/s0305004122000329
{"title":"PSP volume 173 issue 2 Cover and Front matter","authors":"","doi":"10.1017/s0305004122000329","DOIUrl":"https://doi.org/10.1017/s0305004122000329","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"11 1","pages":"f1 - f2"},"PeriodicalIF":0.8,"publicationDate":"2022-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85241319","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 : 2022-08-17DOI: 10.1017/s0305004122000330
{"title":"PSP volume 173 issue 2 Cover and Back matter","authors":"","doi":"10.1017/s0305004122000330","DOIUrl":"https://doi.org/10.1017/s0305004122000330","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"25 1","pages":"b1 - b2"},"PeriodicalIF":0.8,"publicationDate":"2022-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82585825","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 : 2022-08-09DOI: 10.1017/S0305004123000142
Nadav Yesha
Abstract We study intermediate-scale statistics for the fractional parts of the sequence $left(alpha a_{n}right)_{n=1}^{infty}$ , where $left(a_{n}right)_{n=1}^{infty}$ is a positive, real-valued lacunary sequence, and $alphainmathbb{R}$ . In particular, we consider the number of elements $S_{N}!left(L,alpharight)$ in a random interval of length $L/N$ , where $L=O!left(N^{1-epsilon}right)$ , and show that its variance (the number variance) is asymptotic to L with high probability w.r.t. $alpha$ , which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in $alphainmathbb{R}$ when $L=O!left(N^{1/2-epsilon}right)$ . For slowly growing L, we further prove a central limit theorem for $S_{N}!left(L,alpharight)$ which holds for almost all $alphainmathbb{R}$ .
{"title":"Intermediate-scale statistics for real-valued lacunary sequences","authors":"Nadav Yesha","doi":"10.1017/S0305004123000142","DOIUrl":"https://doi.org/10.1017/S0305004123000142","url":null,"abstract":"Abstract We study intermediate-scale statistics for the fractional parts of the sequence \u0000$left(alpha a_{n}right)_{n=1}^{infty}$\u0000 , where \u0000$left(a_{n}right)_{n=1}^{infty}$\u0000 is a positive, real-valued lacunary sequence, and \u0000$alphainmathbb{R}$\u0000 . In particular, we consider the number of elements \u0000$S_{N}!left(L,alpharight)$\u0000 in a random interval of length \u0000$L/N$\u0000 , where \u0000$L=O!left(N^{1-epsilon}right)$\u0000 , and show that its variance (the number variance) is asymptotic to L with high probability w.r.t. \u0000$alpha$\u0000 , which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in \u0000$alphainmathbb{R}$\u0000 when \u0000$L=O!left(N^{1/2-epsilon}right)$\u0000 . For slowly growing L, we further prove a central limit theorem for \u0000$S_{N}!left(L,alpharight)$\u0000 which holds for almost all \u0000$alphainmathbb{R}$\u0000 .","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"107 1","pages":"303 - 318"},"PeriodicalIF":0.8,"publicationDate":"2022-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76243809","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}
$x in (0,1]$ with the following property is comeager: for all integers 
$bge 2$ and 
$kge 1$ , the sequence of vectors made by the frequencies of all possibile strings of length k in the b-adic representation of x has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen (2004) in this Journal. We provide analogues in the context of analytic P-ideals and regular matrices.
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