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MATH SEMINAR:SUMS OF TRIANGULAR NUMBERS AND SUMS OF SQUARES

Speaker: Zafer Selçuk Aygın

Title: SUMS OF TRIANGULAR NUMBERS AND SUMS OF SQUARES

Date/Time: 08 March 2023 / 19:00-20:00

Zoom: Meeting ID:https://sabanciuniv.zoom.us/j/96704489956?pwd Q1NrdnNIOFlrbWFHYlBJNXdmenBCdz09

Passcode: 963957

Abstract:   For non-negative integers a, b, and n, let N(a, b; n) be the number of represen tations of n as a sum of squares with coefficients 1 or 3 (a of ones and b of threes); and let t(a, b; n) be the number of representations of n as a sum of triangular numbers with coefficients 1 or 3 (a of ones and b of threes). By works of Bateman and Knopp & Adiga, Cooper and Han, it is known that for a and b satisfying 1 ≤ a + 3b ≤ 7, we have 

t(a, b; n) = 2  2 + a4+ abN(a, b; 8n + a + 3b)  and for a and b satisfying a + 3b = 8, Baruah, Cooper and Hirschhorn has proven that 

t(a, b; n) = 2  2 + a4+ ab(N(a, b; 8n + a + 3b) − N(a, b; (8n + a + 3b)/4)). Such identities are not known for a + 3b > 8. In this talk we present our recent results (joint with Amir Akbary, University of Lethbridge) on asymptotic equiva lence of formulas similar to the ones above, as n → ∞, for general a and b with a + b even. This extends a result of Bateman, Datskovsky, and Knopp, which was proven via the circle methoxd and use of singular series. We achieve our re sults by using the theory of modular forms to explicitly compute the Eisenstein components of the generating functions of t(a, b; n) and N(a, b; 8n + a + 3b). The method we use is robust and can be adapted in studying the asymptotics of other representation numbers with general coefficients. 

Bio: Zafer Selcuk Aygin obtained his PhD from Carleton University (Ontario, Canada) in 2016. Since then he has held two prestigious postdoctoral fellowships, one at Nanyang Technological University (Singapore) and one at the University of Calgary (Alberta, Canada). Currently he is an Instructor at Northwestern Polytechnic (Alberta, Canada) and an Adjunct Professor at Carleton University (Ontario, Canada). His main research interest is modular forms and their applications in number theory.