Problems in additive number theory: the Goldbach conjecture and its variants
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Salle Fokko du Cloux
ICJ, Université Lyon 1
The talk is devoted to presenting some problems and techniques in additive theory, a specific branch of analytic number theory. More precisely, we begin with an overview of the circle method - a fundamental tool introduced by Hardy, Littlewood and Ramanujan; then, we focus on its application to the Goldbach conjecture, and we underline the role of the so-called density in the study of related additive problems - such as Vinogradov's theorem - compared to this one. In the second part of the seminar, we explore our recent generalizations of two previous results: we first extend to polynomials computed at prime variables, an estimate of M. Cantarini, A. Gambini and A. Zaccagnini (2020) for the average number of representations of an integer as a sum of prime powers. Eventually, starting from the paper of K. Ikeda and A. I. Suriajaya (2025), we move to the arithmetic progressions, and we find the asymptotic behavior for the average number of representations of an integer as a sum of two prime powers and of at least three primes, over multiples of a fixed integer. All of them are joint works with A. Zaccagnini.