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Moments of the Hurwitz zeta function on the critical line Hurwitz函数在临界线上的力矩
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2022-11-28 DOI: 10.1017/S0305004122000457
A. Sahay
{"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}
引用次数: 2
Most numbers are not normal 大多数数字都不正常
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2022-11-28 DOI: 10.1017/s0305004122000469
ANDREA AVENI, PAOLO LEONETTI
{"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}
引用次数: 0
Weil Sums over Small Subgroups 小子群上的和
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2022-11-14 DOI: 10.1017/s0305004123000415
Alina Ostafe, I. Shparlinski, J. Voloch
{"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}
引用次数: 0
Stable finiteness does not imply linear soficity 稳定有限性并不意味着线性稳定性
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2022-10-21 DOI: 10.1017/S0305004123000154
Be'eri Greenfeld
{"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}
引用次数: 0
PSP volume 173 issue 3 Cover and Back matter PSP第173卷第3期封面和封底
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2022-10-19 DOI: 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}
引用次数: 0
PSP volume 173 issue 3 Cover and Front matter PSP第173卷第3期封面和封面问题
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2022-10-19 DOI: 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}
引用次数: 0
Random amenable C*-algebras 随机可服从C*-代数
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2022-10-05 DOI: 10.1017/S0305004123000178
Bhishan Jacelon
{"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}
引用次数: 1
PSP volume 173 issue 2 Cover and Front matter PSP第173卷第2期封面和封面问题
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2022-08-17 DOI: 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}
引用次数: 0
PSP volume 173 issue 2 Cover and Back matter PSP第173卷第2期封面和封底
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2022-08-17 DOI: 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}
引用次数: 0
Intermediate-scale statistics for real-valued lacunary sequences 实值空白序列的中等规模统计量
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2022-08-09 DOI: 10.1017/S0305004123000142
Nadav Yesha
{"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}
引用次数: 2
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