{"title":"Disproving Hooley’s conjecture","authors":"D. Fiorilli, G. Martin","doi":"10.4171/jems/1291","DOIUrl":null,"url":null,"abstract":". Define G ( x ; q ) to be the variance of primes p ≤ x in the arithmetic progressions modulo q , weighted by log p . In analogy with his q -analogue of Selberg’s upper bound on the variance of primes in intervals, Hooley conjectured that as soon as q tends to infinity and x ≥ q , we have the upper bound G ( x ; q ) (cid:28) x log q . This conjecture was proven true over function fields by Keating and Rudnick, using equidistribution results of Katz. In this paper we show that the upper bound does not hold in general, and that G ( x ; q ) can be much larger than x log q for values of q which are (cid:16) log log x . This implies that a conjecture of the first author on the range of validity of Hooley’s conjecture is essentially best possible.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"49 1","pages":""},"PeriodicalIF":2.5000,"publicationDate":"2022-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the European Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/jems/1291","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
. Define G ( x ; q ) to be the variance of primes p ≤ x in the arithmetic progressions modulo q , weighted by log p . In analogy with his q -analogue of Selberg’s upper bound on the variance of primes in intervals, Hooley conjectured that as soon as q tends to infinity and x ≥ q , we have the upper bound G ( x ; q ) (cid:28) x log q . This conjecture was proven true over function fields by Keating and Rudnick, using equidistribution results of Katz. In this paper we show that the upper bound does not hold in general, and that G ( x ; q ) can be much larger than x log q for values of q which are (cid:16) log log x . This implies that a conjecture of the first author on the range of validity of Hooley’s conjecture is essentially best possible.
期刊介绍:
The Journal of the European Mathematical Society (JEMS) is the official journal of the EMS.
The Society, founded in 1990, works at promoting joint scientific efforts between the many different structures that characterize European mathematics. JEMS will publish research articles in all active areas of pure and applied mathematics. These will be selected by a distinguished, international board of editors for their outstanding quality and interest, according to the highest international standards.
Occasionally, substantial survey papers on topics of exceptional interest will also be published. Starting in 1999, the Journal was published by Springer-Verlag until the end of 2003. Since 2004 it is published by the EMS Publishing House. The first Editor-in-Chief of the Journal was J. Jost, succeeded by H. Brezis in 2004.
The Journal of the European Mathematical Society is covered in:
Mathematical Reviews (MR), Current Mathematical Publications (CMP), MathSciNet, Zentralblatt für Mathematik, Zentralblatt MATH Database, Science Citation Index (SCI), Science Citation Index Expanded (SCIE), CompuMath Citation Index (CMCI), Current Contents/Physical, Chemical & Earth Sciences (CC/PC&ES), ISI Alerting Services, Journal Citation Reports/Science Edition, Web of Science.