A Novel Integral Equation for the Riemann Zeta Function and Large <i>t</i>-Asymptotics

Based on the new approach to Lindel&#246;f hypothesis recently introduced by one of the authors, we first derive a novel integral equation for the square of the absolute value of the Riemann zeta function. Then, we introduce the machinery needed to obtain an estimate for the solution of this equ...

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Bibliographic Details
Published in:Mathematics
Main Authors: Konstantinos Kalimeris, Athanassios S. Fokas
Format: Article
Language:English
Published: MDPI AG 2019-07-01
Subjects:
riemann zeta function
asymptotics
exponential sums
Online Access:https://www.mdpi.com/2227-7390/7/7/650
Description
Summary:Based on the new approach to Lindel&#246;f hypothesis recently introduced by one of the authors, we first derive a novel integral equation for the square of the absolute value of the Riemann zeta function. Then, we introduce the machinery needed to obtain an estimate for the solution of this equation. This approach suggests a substantial improvement of the current large <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>-</mo> </mrow> </semantics> </math> </inline-formula>asymptotics estimate for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#950;</mi> <mfenced separators="" open="(" close=")"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>+</mo> <mi>i</mi> <mi>t</mi> </mfenced> </mrow> </semantics> </math> </inline-formula>.
ISSN:2227-7390

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