Automated Discovery of New $L$-Function Relations

Abstract

$L$-functions typically encode interesting information about mathematical objects. This paper reports 29 identities between such functions that hitherto never appeared in the literature. Of these we have a complete proof for 9; all others are extensively numerically checked and we welcome proofs of their (in)validity. The method we devised to obtain these identities is a two-step process whereby a list of candidate identities is automatically generated, obtained, tested, and ultimately formally proven. The approach is however only \emph{semi-}automated as human intervention is necessary for the post-processing phase, to determine the most general form of a conjectured identity and to provide a proof for them. This work complements other instances in the literature where automated symbolic computation has served as a productive step toward theorem proving and can be extended in several directions further to explore the algebraic landscape of $L$-functions and similar constructions.