Formalizing an analytic proof of the Prime Number Theorem (extended abstract) Dedicated to Mike Gordon on the occasion of his 60th birthday

1 Formalizing mathematics: pure and applied I've always been interested in using theorem provers both for " practical " applications in formally verifying computer systems, and for the " pure " formalization of traditional mathematical proofs. I particularly like situations where there is an interplay between the two. For example, in my PhD thesis [5], written under Mike Gordon's supervision, I developed a formalization of some elementary real analysis. This was subsequently used in very practical verification applications [6], where in fact I even needed to formalize more pure mathematics, such as power series for the cotangent function and basic theorems about diophantine approximation. I first joined Mike Gordon's HVG (Hardware Verification Group) to work on an embedding in HOL of the hardware description language ELLA. Mike had already directed several similar research projects, and one concept first clearly articulated as a result of these activities was the now-standard distinction between 'deep' and 'shallow' embeddings of languages [3]. Since I was interested in formalizing real analysis, Mike encouraged me to direct my attention to case studies involving arithmetic, and this was the starting-point for my subsequent research. Right from the beginning, Mike was very enthusiastic about my formalization of the reals from first principles using Dedekind cuts. Mike had been involved in Robin Milner's group developing the original Edinburgh LCF [4], a central feature of which was the idea of extending the logical basis with derived inference rules to preserve soundness. Now that Mike had applied the LCF approach to higher-order logic, suitable as a general foundation for mathematics, it was possible to extend this idea and even develop mathematical concepts themselves in a 'correct by construction' way using definitions. So a definitional construction of the reals fitted in very well with the ideals Mike had for the HOL project, an interest in applications combined with an emphasis on careful foundations that has now become commonplace. In this paper I want to describe a formalization that was undertaken purely for fun, involving complex analysis [8] and culminating in a proof of the Prime Number Theorem. Nevertheless, it doesn't seem entirely far-fetched to imagine some " practical " applications of this result in the future. For example a weak form of the PNT is implicitly used to justify the termination of the breakthrough AKS primality test [1], and