{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:35:37Z","timestamp":1776785737739,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"254","license":[{"start":{"date-parts":[[2006,11,30]],"date-time":"2006-11-30T00:00:00Z","timestamp":1164844800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Each simple zero\n \n \n \n \n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n +<\/mml:mo>\n i<\/mml:mi>\n \n \n \u03b3\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n \\frac {1}{2}+i\\gamma _n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the Riemann zeta function on the critical line with\n \n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\gamma _n > 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a center for the flow\n \n \n \n \n \n \n s<\/mml:mi>\n \n \u02d9\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n =<\/mml:mo>\n \n \u03be\n \n <\/mml:mi>\n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\dot {s}=\\xi (s)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the Riemann xi function with an associated period\n \n \n \n \n T<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n T_n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . It is shown that, as\n \n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n \u2192\n \n <\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n \\gamma _n \\rightarrow \\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \\[\n \n \n \n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n T<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n \u2265\n \n <\/mml:mo>\n \n \n \u03c0\n \n <\/mml:mi>\n 4<\/mml:mn>\n <\/mml:mfrac>\n \n \n \u03b3\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n +<\/mml:mo>\n O<\/mml:mi>\n (<\/mml:mo>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n \n \u03b3\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n .<\/mml:mo>\n <\/mml:mrow>\n \\log T_n\\ge \\frac {\\pi }{4}\\gamma _n+O(\\log \\gamma _n).<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n Numerical evaluation leads to the conjecture that this inequality can be replaced by an equality. Assuming the Riemann Hypothesis and a zeta zero separation conjecture\n \n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n \n n<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2212\n \n <\/mml:mo>\n \n \n \u03b3\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n \u226b\n \n <\/mml:mo>\n \n \n \u03b3\n \n <\/mml:mi>\n n<\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n \n \u03b8\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n \\gamma _{n+1}-\\gamma _n \\gg \\gamma _n^{-\\theta }<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for some exponent\n \n \n \n \n \n \u03b8\n \n <\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\theta >0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , we obtain the upper bound\n \n \n \n \n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n T<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n \u226a\n \n <\/mml:mo>\n \n \n \u03b3\n \n <\/mml:mi>\n n<\/mml:mi>\n \n 2<\/mml:mn>\n +<\/mml:mo>\n \n \u03b8\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n \\log T_n \\ll \\gamma ^{2+\\theta }_n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Assuming a weakened form of a conjecture of Gonek, giving a bound for the reciprocal of the derivative of zeta at each zero, we obtain the expected upper bound for the periods so, conditionally,\n \n \n \n \n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n T<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n \n \n \u03c0\n \n <\/mml:mi>\n 4<\/mml:mn>\n <\/mml:mfrac>\n \n \n \u03b3\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n +<\/mml:mo>\n O<\/mml:mi>\n (<\/mml:mo>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n \n \u03b3\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \\log T_n = \\frac {\\pi }{4}\\gamma _n+O(\\log \\gamma _n)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Indeed, this linear relationship is equivalent to the given weakened conjecture, which implies the zero separation conjecture, provided the exponent is sufficiently large. The frequencies corresponding to the periods relate to natural eigenvalues for the Hilbert\u2013Polya conjecture. They may provide a goal for those seeking a self-adjoint operator related to the Riemann hypothesis.\n <\/p>","DOI":"10.1090\/s0025-5718-05-01803-x","type":"journal-article","created":{"date-parts":[[2006,2,15]],"date-time":"2006-02-15T11:05:20Z","timestamp":1140001520000},"page":"891-902","source":"Crossref","is-referenced-by-count":2,"title":["Linear law for the logarithms of the Riemann periods at simple critical zeta zeros"],"prefix":"10.1090","volume":"75","author":[{"given":"Kevin","family":"Broughan","sequence":"first","affiliation":[]},{"given":"A.","family":"Barnett","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,11,30]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"236","DOI":"10.1137\/S0036144598347497","article-title":"The Riemann zeros and eigenvalue asymptotics","volume":"41","author":"Berry, M. 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M.","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0486417409"},{"issue":"10","key":"9","first-page":"741","article-title":"On the difference between \ud835\udc5f consecutive ordinates of the zeros of the Riemann zeta function","volume":"51","author":"Fujii, Akio","year":"1975","journal-title":"Proc. Japan Acad.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-4280","issn-type":"print"},{"key":"10","unstructured":"Godfrey, P. An efficient algorithm for the Riemann zeta function, \\url{http:\/\/www.mathworks.com\/support\/ftp} zeta.m, etan.m (2000)"},{"key":"11","unstructured":"Gonek, S.M. The second moment of the reciprocal of the Riemann zeta function and its derivative, lecture at the Mathematical Sciences Research Institute, Berkeley (June 1999)."},{"issue":"2003","key":"12","doi-asserted-by":"publisher","first-page":"2611","DOI":"10.1098\/rspa.2000.0628","article-title":"Random matrix theory and the derivative of the Riemann zeta function","volume":"456","author":"Hughes, C. 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