{"title":"欧拉定理的分布模p与真除数和","authors":"Noah Lebowitz-Lockard, P. Pollack, A. Roy","doi":"10.1307/mmj/20216082","DOIUrl":null,"url":null,"abstract":"Abstract. We consider the distribution in residue classes modulo primes p of Euler’s totient function φ(n) and the sum-of-proper-divisors function s(n) := σ(n)−n. We prove that the values φ(n), for n ≤ x, that are coprime to p are asymptotically uniformly distributed among the p−1 coprime residue classes modulo p, uniformly for 5 ≤ p ≤ (log x) (with A fixed but arbitrary). We also show that the values of s(n), for n composite, are uniformly distributed among all p residue classes modulo every p ≤ (log x). These appear to be the first results of their kind where the modulus is allowed to grow substantially with x.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"24 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Distribution mod p of Euler’s Totient and the Sum of Proper Divisors\",\"authors\":\"Noah Lebowitz-Lockard, P. Pollack, A. Roy\",\"doi\":\"10.1307/mmj/20216082\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract. We consider the distribution in residue classes modulo primes p of Euler’s totient function φ(n) and the sum-of-proper-divisors function s(n) := σ(n)−n. We prove that the values φ(n), for n ≤ x, that are coprime to p are asymptotically uniformly distributed among the p−1 coprime residue classes modulo p, uniformly for 5 ≤ p ≤ (log x) (with A fixed but arbitrary). We also show that the values of s(n), for n composite, are uniformly distributed among all p residue classes modulo every p ≤ (log x). These appear to be the first results of their kind where the modulus is allowed to grow substantially with x.\",\"PeriodicalId\":49820,\"journal\":{\"name\":\"Michigan Mathematical Journal\",\"volume\":\"24 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2021-05-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Michigan Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1307/mmj/20216082\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Michigan Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1307/mmj/20216082","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Distribution mod p of Euler’s Totient and the Sum of Proper Divisors
Abstract. We consider the distribution in residue classes modulo primes p of Euler’s totient function φ(n) and the sum-of-proper-divisors function s(n) := σ(n)−n. We prove that the values φ(n), for n ≤ x, that are coprime to p are asymptotically uniformly distributed among the p−1 coprime residue classes modulo p, uniformly for 5 ≤ p ≤ (log x) (with A fixed but arbitrary). We also show that the values of s(n), for n composite, are uniformly distributed among all p residue classes modulo every p ≤ (log x). These appear to be the first results of their kind where the modulus is allowed to grow substantially with x.
期刊介绍:
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