Paper 2021/1060

Discovering New $L$-Function Relations Using Algebraic Sieving

Hadrien Barral, Éric Brier, Rémi Géraud-Stewart, Arthur Léonard, David Naccache, Quentin Vermande, and Samuel Vivien

Abstract

We report the discovery of new results relating $L$-functions, which typically encode interesting information about mathematical objects, obtained in a \emph{semi-automated} fashion using an algebraic sieving technique. Algebraic sieving initially comes from cryptanalysis, where it is used to solve factorization, discrete logarithms, or to produce signature forgeries in cryptosystems such as RSA. We repurpose the technique here to provide candidate identities, which can be tested and ultimately formally proven. A limitation of our technique is the need for human intervention in the post-processing phase, to determine the most general form of conjectured identities, and to provide a proof for them. Nevertheless we report 29 identities that hitherto never appeared in the literature, 9 of which we could completely prove, the remainder being numerically valid over all tested values. This work complements other instances in the literature where this type of automated symbolic computation has served as a productive step toward theorem proving; it can be extremely helpful in figuring out what it is that one should attempt to prove.