{"docId":155,"paperId":155,"url":"https:\/\/hrj.episciences.org\/155","doi":"10.46298\/hrj.2006.155","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":40,"name":"Volume 29 - 2006"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01111487","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-01111487v1","dateSubmitted":"2015-03-03 16:13:58","dateAccepted":"2015-06-12 16:05:51","datePublished":"2006-01-01 08:00:00","titles":{"en":"A remark on a theorem of A. E. Ingham."},"authors":["Bhat, K G","Ramachandra, K"],"abstracts":{"en":"Referring to a theorem of A. E. Ingham, that for all $N\\geq N_0$ (an absolute constant), the inequality $N^3\\leq p\\leq(N+1)^3$ is solvable in a prime $p$, we point out in this paper that it is implicit that he has actually proved that $\\pi(x+h)-\\pi(x) \\sim h(\\log x)^{-1}$ where $h=x^c$ and $c (>\\frac{5}{8})$ is any constant. Further, we point out that even this stronger form can be proved without using the functional equation of $\\zeta(s)$."},"keywords":[{"en":" Littlewood"},{"en":" Weyl"},{"en":" Hurwitz Zeta-function"},{"en":" Hardy"},{"en":" Riemann Zeta-function"},{"en":"Analytic Number Theory"},"[MATH] Mathematics [math]"]}