A survey of Büthe's method for estimating prime counting functions

Abstract

This thesis provides explicit bounds for the Chebyshev prime counting function ψ(x). This thesis aims to produce a detailed survey of the first part (from page 2483 to page 2494) of the paper, ‘Estimating π(x) and Related Functions Under Partial RH Assumptions’ by Jan B ̈uthe published in 2016. His article provides the best-known bounds for ψ(x) for x ≤ e3000 using the Fourier Transform of the Logan Function, assuming the Riemann Hypothesis to be valid for all zeroes of the zeta function with ℑ(ρ) ∈ (0, T ] for a specific T. The main theorem in B ̈uthe’s paper gives a bound for |ψ(x) − x| using an equation with three major terms E1, E2 and E3 and provides bounds for each of these terms individually. The necessary lemmas, propositions and their proofs required to prove the main theorem are scattered throughout various papers such as [7], [8], [10], [18], and [6]. In this thesis, we have accumulated all these results, verified their proofs, and included various missing details. Several of the arguments in the original paper have been reworked, and necessary corrections, such as rectifying the error terms E2 and E3 in the main theorem, and other minor amendments have been made with the goal of turning this thesis into a self-contained research exposition of B ̈uthe’s work in [7].