Zeta Types and Tannakian Symbols as a Method for Representing Mathematical Knowledge
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We present two closely related notions called (1) a zeta type and (2) a Tannakian symbol. These are data types for representing information about number-theoretic objects, and we argue that a database built out of zeta types and Tannakian symbols could lead to interesting discoveries, similar to what has been achieved for example by the OEIS, the LMFDB, and other existing databases of mathematical objects. We give several examples illustrating what database records would look like, and we describe a tiny prototype database which has already been used to find and automatically prove new theorems about multiplicative functions.
KeywordsNumber theory Multiplicative functions Tannakian symbols Zeta types Zeta functions L-functions Automated conjecture-making Automated theorem proving
Aspects of this work has been presented in various talks, including at CICM 2016, at AITP 2017 (Conference on Artificial Intelligence and Theorem Proving), at the Representation Theory 2016 Conference in Uppsala, and at seminars in Stockholm and in Leicester. We would like to acknowledge the encouragement and many helpful comments received on these occasions, and in particular we thank Michael Kohlhase, Volodymyr Mazorchuk, Frank Neumann, Florian Rabe, Andreas Strömbergsson, and Josef Urban.
(Slides from some of these talks are available on the first author’s webpage; andreasholmstrom.org. Together with the other documents mentioned, they may serve as a complement to this paper.)
Finally, we also want to thank Magnus Hellebust Haaland and Olav Hellebust Haaland for implementing the Berlekamp-Massey algorithm for us in Sage; many of the automated procedures for computing with Tannakian symbols rely on this implementation.
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