Abstract
In 2009 Fokas began a program of study of the investigation of the large t-asymptotics of the Riemann zeta function, \(\zeta (\sigma +it)\). In the current work we present a novel difference-integral equation which is satisfied asymptotically by \(\zeta (1/2+it)\). This equation is obtained starting with a singular integral equation presented for the first time in 2019 and using a finite Fourier transform representation of the Riemann zeta function. The relevant analysis involves a plethora of tools and techniques developed by Fokas and collaborators during the last decade.
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References
Erdélyi, A.: Asymptotic Expansions (No. 3). Courier Corporation (1956)
Fokas, A.S.: A unified transform method for solving linear and certain nonlinear PDEs. Proc. R. Soc. Lond. Ser. Math. Phys. Eng. Sci. 453(1962), 1411–1443 (1997)
Fokas, A.S.: A novel approach to the Lindelöf hypothesis. Trans. Math. Appl. 3, 1–49 (2019)
Fokas, A.S., Kaxiras, E.: Modern Mathematical Methods for Scientists and Engineers: A Street-Smart Introduction. World Scientific (2022)
Fokas, A.S., Lenells, J.: On the asymptotics to all orders of the Riemann Zeta function and of a two-parameter generalization of the Riemann Zeta function. Mem. AMS. 275(1351) (2022)
Kalimeris, K., Fokas, A.S.: Explicit asymptotics for certain single and double exponential sums. Proc. R. Soc. Edinb. Math. 150(2), 607–632 (2020)
Kalimeris, K., Fokas, A.S.: A novel integral equation for the Riemann zeta function and large t-asymptotics. Math. 7(7), 650 (2019)
Siegel, C.L., Nachlaßzur, Über Riemanns, analytischen Zahlentheorie, Quellen Studien zur Geschichte der Math. Astron. und Phys. Abt. B: Studien 2: 4580,: reprinted in Gesammelte Abhandlungen, vol. 1, p. 1966. Springer-Verlag, Berlin (1932)
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Fokas, A.S., Kalimeris, K., Lenells, J. (2023). A Novel Difference-Integral Equation Satisfied Asymptotically by the Riemann Zeta Function. In: Bountis, T., Vallianatos, F., Provata, A., Kugiumtzis, D., Kominis, Y. (eds) Chaos, Fractals and Complexity. COSA-Net 2022. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-031-37404-3_22
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