{"docId":133,"paperId":133,"url":"https:\/\/hrj.episciences.org\/133","doi":"10.46298\/hrj.1996.133","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":30,"name":"Volume 19 - 1996"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01109304","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-01109304v1","dateSubmitted":"2015-03-03 16:13:50","dateAccepted":"2015-06-12 16:05:38","datePublished":"1996-01-01 08:00:00","titles":{"en":"Ramanujan's lattice point problem, prime number theory and other remarks."},"authors":["Ramachandra, K","Sankaranarayanan, A","Srinivas, K"],"abstracts":{"en":"This paper gives results on four diverse topics. The first result is that the error term for the number of integers $2^u3^v \\le n$ is $O((\\log n)^{1-\\delta})$ with $\\delta=(2^{40}(\\log3))^{-1}$, using a theorem of A. Baker and G. W\\\"ustholz. The second result is an averaged explicit formula \\[ \\psi(x) = x-\\frac{1}{T} \\int_{T}^{2T} \\left( \\sum \\limits_{|\\gamma| \\le \\tau} \\frac{x^{\\rho}}{\\rho} \\right) \\ d\\tau + O \\left( \\frac{\\log x}{\\log \\frac{x}{T}}\\cdot \\frac{x}{T} \\right) \\] for $x \\gg T \\gg 1$. It then follows, by the Riemann hypothesis, that $\\psi (x+h)-\\psi (x)= h+ O \\left ( h \\lambda^{1\/2} \\right )$ if $h=\\lambda x^{1\/2} \\log x$. The third theme tightens the $\\log$ powers in the zero density bounds of Ingham and Huxley, and gives corollaries for the mean-value of $\\psi (x+h)-\\psi (x)-h$. The fourth remark concerns a hypothetical improvement in the constant 2 in the Brun-Titchmarsh theorem, averaged over congruence classes, and its consequence for $L \\left ( 1,\\chi \\right )$."},"keywords":[{"en":"average explicit formula"},{"en":" zero-density bounds"},{"en":" Brun-Titchmarsh Theorem"},"[MATH] Mathematics [math]"]}