{"title":"The Carlson-type zero-density theorem for the Beurling \u0000 \u0000 ζ\u0000 $zeta$\u0000 function","authors":"Szilárd Gy. Révész","doi":"10.1112/jlms.70110","DOIUrl":"https://doi.org/10.1112/jlms.70110","url":null,"abstract":"<p>In a previous paper, we proved a Carlson-type density theorem for zeroes in the critical strip for the Beurling zeta functions satisfying Axiom A of Knopfmacher. There we needed to invoke two additional conditions: the integrality of the norm (Condition B) and an “average Ramanujan condition” for the arithmetical function counting the number of different Beurling integers of the same norm <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>∈</mo>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$min {mathbb {N}}$</annotation>\u0000 </semantics></math> (Condition G).</p><p>Here, we implement a new approach of Pintz using the classic zero-detecting sums coupled with Halász' method, but otherwise arguing in an elementary way avoiding, for example, large sieve-type inequalities or mean value estimates for Dirichlet polynomials. In this way, we give a new proof of a Carlson-type density estimate—with explicit constants—avoiding any use of the two additional conditions needed earlier.</p><p>Therefore, it is seen that the validity of a Carlson-type density estimate does not depend on any extra assumption—neither on the functional equation present for the Selberg class, nor on growth estimates of coefficients say of “average Ramanujan-type”—but is a general property presenting itself whenever the analytic continuation is guaranteed by Axiom A.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70110","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143554362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}