Pub Date : 2023-02-08DOI: 10.1017/s0305004123000026
{"title":"PSP volume 174 issue 2 Cover and Back matter","authors":"","doi":"10.1017/s0305004123000026","DOIUrl":"https://doi.org/10.1017/s0305004123000026","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90626846","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 : 2023-02-08DOI: 10.1017/s0305004123000014
{"title":"PSP volume 174 issue 2 Cover and Front matter","authors":"","doi":"10.1017/s0305004123000014","DOIUrl":"https://doi.org/10.1017/s0305004123000014","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91000428","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 : 2022-12-19DOI: 10.1017/s0305004122000494
{"title":"PSP volume 174 issue 1 Cover and Front matter","authors":"","doi":"10.1017/s0305004122000494","DOIUrl":"https://doi.org/10.1017/s0305004122000494","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77604410","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 : 2022-12-19DOI: 10.1017/s0305004122000500
{"title":"PSP volume 174 issue 1 Cover and Back matter","authors":"","doi":"10.1017/s0305004122000500","DOIUrl":"https://doi.org/10.1017/s0305004122000500","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76545348","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/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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
$x in (0,1]$ 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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