{"title":"Large gaps between primes","authors":"Kevin Ford","doi":"10.1090/gsm/203/30","DOIUrl":"https://doi.org/10.1090/gsm/203/30","url":null,"abstract":"denote the the maximum gap between consecutive primes less than X . It is clear from the prime number theorem that G(X) > (1 + o(1)) logX, as the average gap between the prime numbers which are 6 X is ∼ logX . Explicit gaps of (at least) this size may be constructed by observing that P (n)+2, P (n)+3, . . . , P (n)+n is a sequence of n−1 composite numbers, where P (n) is the product of the primes 6 n; by the prime number theorem, P (n) = e(1+o(1))n. The decade of the 1930s witnessed a lot of improvements to bounds for G(X). First, Backlund [1] obtained G(X) > (2 + o(1)) logX in 1930, and Brauer-Zeitz [3] got G(X) > (4 + o(1)) logX in 1931. The same year, Westzynthius [25] obtained the first order of magnitude improvement, and consequently proved that infinitely often the gap between consecutive prime numbers can be an arbitrarily large multiple of the average gap. His bound is","PeriodicalId":140374,"journal":{"name":"Graduate Studies in Mathematics","volume":"105 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114408385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Irregularities in the distribution of\u0000 primes","authors":"","doi":"10.1090/gsm/203/31","DOIUrl":"https://doi.org/10.1090/gsm/203/31","url":null,"abstract":"","PeriodicalId":140374,"journal":{"name":"Graduate Studies in Mathematics","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127700966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sieving for zero-free regions","authors":"","doi":"10.1090/gsm/203/23","DOIUrl":"https://doi.org/10.1090/gsm/203/23","url":null,"abstract":"","PeriodicalId":140374,"journal":{"name":"Graduate Studies in Mathematics","volume":"192 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114746652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Distribution of Prime Numbers","authors":"Dimitris Koukoulopoulos","doi":"10.1090/gsm/203","DOIUrl":"https://doi.org/10.1090/gsm/203","url":null,"abstract":"","PeriodicalId":140374,"journal":{"name":"Graduate Studies in Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122685364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Prime Number Theorem for arithmetic\u0000 progressions","authors":"","doi":"10.1090/gsm/203/13","DOIUrl":"https://doi.org/10.1090/gsm/203/13","url":null,"abstract":"","PeriodicalId":140374,"journal":{"name":"Graduate Studies in Mathematics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132535866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Small gaps between primes","authors":"A. Mocanu","doi":"10.1090/gsm/203/29","DOIUrl":"https://doi.org/10.1090/gsm/203/29","url":null,"abstract":"In this paper we study the work of James Maynard [10], in which he proves that lim inf n→∞ (pn+m − pn) <∞, thus establishing that there are infinitely many intervals of finite length that contain a fixed number of primes. We use Sieve theory, in particular Selberg’s sieve, and we look at preliminary work in order to better understand this result. To conclude, we discuss the implications of this result.","PeriodicalId":140374,"journal":{"name":"Graduate Studies in Mathematics","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134354189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Riemann zeta function","authors":"Sebastián Carrillo Santana","doi":"10.1090/gsm/203/07","DOIUrl":"https://doi.org/10.1090/gsm/203/07","url":null,"abstract":"The purpose of this thesis is to give a brief introduction to the Riemann Zeta function, explaining the most important properties such as the analytic continuation, the functional equation and its zeros.","PeriodicalId":140374,"journal":{"name":"Graduate Studies in Mathematics","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129620839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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