Nonstandard Mathematics and New Zeta and L-Functions

TL;DR

This thesis introduces new nonstandard zeta functions, explores their analytical properties, and develops p-adic interpolation for multiple primes, offering a novel perspective on existing mathematical frameworks.

Contribution

It defines new nonstandard zeta functions, develops their analysis, and extends p-adic interpolation to multiple primes, connecting to Haran's work from a nonstandard viewpoint.

Findings

Defined new nonstandard zeta functions with analytical properties

Extended p-adic interpolation to multiple primes

Provided a nonstandard perspective on Haran's work

Abstract

This Ph.D. thesis, prepared under the supervision of Professor Ivan Fesenko, defines new zeta functions in a nonstandard setting and their analytical properties are developed. Further, p-adic interpolation is presented within a nonstandard setting which enables the concept of interpolating with respect to two, or more, distinct primes to be defined. The final part of the dissertation examines the work of M. J. Shai Haran and makes initial attempts of viewing it from a nonstandard perspective.