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DTSTART;TZID=America/New_York:20260922T140000
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DTSTAMP:20260731T003059
CREATED:20260630T154021Z
LAST-MODIFIED:20260728T155040Z
UID:10003954-1790085600-1790089200@cmsa.fas.harvard.edu
SUMMARY:CMSA Special Seminar: Deformations to the Complex Plane\, Novel Asymptotic Techniques\, and the Large t- Asymptotics of the Riemann Zeta Function
DESCRIPTION:CMSA Special Seminar \n \nSpeaker: Thanasis Fokas\, University of Cambridge \nTitle: Deformations to the Complex Plane\, Novel Asymptotic Techniques\, and the Large t-\nAsymptotics of the Riemann Zeta Function \nAbstract: The Unified Transform (also known as the Fokas method) is a powerful new method for\nsolving boundary value problems for linear and for integrable nonlinear PDEs. For linear\nPDEs\, the relevant transform is based on appropriate deformations of certain integrals from\nthe real line to the complex plane. After briefly reviewing this transform\, it will be shown\nthat combining this idea with novel asymptotic techniques has recently led to unexpected\nand exciting results regarding the large t-asymptotic analysis of the celebrated Riemann zeta\nfunction. The following two results will be discussed. First\, a simple formula will be\npresented for the difference of the functions defining the error terms in two historic\nproblems: in Atkinson’s formula and in the formula for the Dirichlet divisor problem; it will\nbe shown that this difference equals π/ 2 plus a function which is simply related to the\nsquare of the Riemann zeta function. Second\, a remarkable integral identity satisfied by the\nRiemann zeta function will be presented; this identity is obtained from an earlier identity\nderived by the speaker via contour deformation in the complex plane. Making crucial use of\nnovel asymptotic techniques obtained jointly with Jonatan Lenells\, the asymptotic analysis\nof this integral equation gives rise to an interesting identity satisfied\, for large t\, by a sum\ngeneralizing the Dirichlet divisor sum. Also\, and more importantly\, it gives rise to a specific\nintegral transform suitable for the large t-asymptotic analysis of the Riemann zeta function. \n 
URL:https://cmsa.fas.harvard.edu/event/fokas/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquia & Seminar,Special Seminar
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Special-Seminar-9.22.26.docx-scaled.png
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