A Survey of Büthe's Method for Estimating Prime Counting Functions

Abstract: This talk will begin with a study on explicit bounds for $\psi(x)$ starting with the work of Rosser in 1941. It will also cover various improvements over the years including the works of Rosser and Schoenfeld, Dusart, Faber-Kadiri, Platt-Trudgian, Büthe, and Fiori-Kadiri-Swidinsky. In the second part of this talk, I will provide an overview of my master's thesis which is a survey on `Estimating $\pi(x)$ and Related Functions under Partial RH Assumptions' by Jan Büthe. This article provides the best known bounds for $\psi(x)$ for small values of~$x$ in the interval $[e^{50},e^{3000}]$. A distinctive feature of this paper is the use of Logan's function and its Fourier Transform. I will be presenting the main theorem in Büthe's paper regarding estimates for $\psi(x)$ with other necessary results required to understand the proof.

combinatoricsnumber theory

Audience: researchers in the topic

Lethbridge number theory and combinatorics seminar