{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,2]],"date-time":"2025-12-02T22:15:42Z","timestamp":1764713742895},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"351","license":[{"start":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T00:00:00Z","timestamp":1749168000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS 2101912","712898"],"award-info":[{"award-number":["DMS 2101912","712898"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>Assuming the Riemann hypothesis, we improve the current upper and lower bounds for the average value of Montgomery\u2019s function <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F left-parenthesis alpha comma upper T right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>F<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>\u03b1<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>T<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">F(\\alpha , T)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> over long intervals by means of a Fourier optimization framework. The function <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F left-parenthesis alpha comma upper T right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>F<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>\u03b1<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>T<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">F(\\alpha , T)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is often used to study the pair correlation of the non-trivial zeros of the Riemann zeta-function. Two ideas play a central role in our approach: (i) the introduction of new averaging mechanisms in our conceptual framework and (ii) the full use of the class of test functions introduced by Cohn and Elkies for the sphere packing bounds, going beyond the usual class of bandlimited functions. We conclude that such an average value, that is conjectured to be <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1\">\n  <mml:semantics>\n    <mml:mn>1<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, lies between <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0.9303\">\n  <mml:semantics>\n    <mml:mn>0.9303<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">0.9303<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1.3208\">\n  <mml:semantics>\n    <mml:mn>1.3208<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">1.3208<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. Our Fourier optimization framework also yields an improvement on the current bounds for the analogous problem concerning the non-trivial zeros in the family of Dirichlet <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L\">\n  <mml:semantics>\n    <mml:mi>L<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">L<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-functions.<\/p>","DOI":"10.1090\/mcom\/3990","type":"journal-article","created":{"date-parts":[[2024,5,11]],"date-time":"2024-05-11T02:00:48Z","timestamp":1715392848000},"page":"409-424","source":"Crossref","is-referenced-by-count":1,"title":["Fourier optimization and Montgomery\u2019s pair correlation conjecture"],"prefix":"10.1090","volume":"94","author":[{"given":"Emanuel","family":"Carneiro","sequence":"first","affiliation":[]},{"given":"Micah","family":"Milinovich","sequence":"additional","affiliation":[]},{"given":"Antonio Pedro","family":"Ramos","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2024,6,6]]},"reference":[{"key":"1","unstructured":"S. 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