Abstract
This is a brief summary of a seminar talk, with the same title, delivered at the CRM—Barcelona in October, 2019. We discuss here some of the recent bounds for objects related to the Riemann zeta-function and prime gaps via the use of Fourier analysis machinery. Certain interesting Fourier optimization problems come into play, naturally related to our number theoretical entities. This is based on a joint work with M. Milinovich and K. Soundararajan.
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Notes
- 1.
In fact, this was our initial approach for the result that later became [4, Theorem 4].
References
N.N. Andreev, S.V. Konyagin, A.Y. Popov, Extremum problems for functions with small support. Math. Notes 60(3) (1996). (translated from Mat. Zametki)
E. Carneiro, V. Chandee, M.B. Milinovich, Bounding \(S(t)\) and \(S_1(t)\) on the Riemann hypothesis. Math. Ann. 356(3), 939–968 (2013)
E. Carneiro, A. Chirre, Bounding \(S_n(t)\) on the Riemann hypothesis. Math. Proc. Camb. Philos. Soc. 164(2), 259–283 (2018)
E. Carneiro, M. Milinovich, K. Soundararajan, Fourier optimization and prime gaps. Comment. Math. Helv. 94(3), 533–568 (2019)
V. Chandee, K. Soundararajan, Bounding \(|\zeta (\frac{1}{2}+it)|\) on the Riemann hypothesis. Bull. Lond. Math. Soc. 43(2), 243–250 (2011)
H. Cramér, Some theorems concerning prime numbers. Ark. Mat. Astron. Fysik. 15(5), 1–32 (1920)
H. Cohn, N. Elkies, New upper bounds on sphere packings. I. Ann. Math. (2) 157(2), 689–714 (2003)
H. Cohn, A. Kumar, S.D. Miller, D. Radchenko, M. Viazovska, The sphere packing problem in dimension 24. Ann. Math. 185, 1017–1033 (2017)
A. Dudek, On the Riemann hypothesis and the difference between primes. Int. J. Num. Theory 11(3), 771–778 (2015)
D.A. Goldston, On a result of Littlewood concerning prime numbers, Acta Arith. 40(3), 263–271 (1981/82)
D.V. Gorbachev, An integral problem of Konyagin and the (C, L)-constants of Nikol’skii. Trudy Inst. Mat. i Mekh. UrO RAN 11(2), 72–91 (2005)
L. Hörmander, B. Bernhardsson, An extension of Bohr’s inequality. Boundary value problems for partial differential equations and applications, 179–194, RMA Res. Notes Appl. Math., 29, Masson, Paris (1993)
H. Iwaniec, E. Kowalski, Analytic Number Theory. Am. Math. Soc. Colloq. Publ. 53(2004)
M. Plancherel, G. Pólya, Fonctions entiéres et intégrales de Fourier multiples, (Seconde partie) Comment. Math. Helv. 10, 110–163 (1938)
O. Ramaré, Y. Saouter, Short effective intervals containing primes. J. Num. Theory 98(1), 10–33 (2003)
J.D. Vaaler, Some extremal functions in Fourier analysis. Bull. Am. Math. Soc. 12, 183–215 (1985)
M. Viazovska, The sphere packing problem in dimension 8. Ann. Math. 185, 991–1015 (2017)
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Carneiro, E. (2021). Bounds for Zeta and Primes via Fourier Analysis. In: Abakumov, E., Baranov, A., Borichev, A., Fedorovskiy, K., Ortega-Cerdà, J. (eds) Extended Abstracts Fall 2019. Trends in Mathematics(), vol 12. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-74417-5_9
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