Towards The Siegel-Walfisz Theorem

Deb, Nripendra Kumar (2025) Towards The Siegel-Walfisz Theorem. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Nripendra Kumar Deb (20MS198)) 20MS198_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (540kB)

Abstract

This thesis traces the path from classical results in analytic number theory to the Siegel–Walfisz theorem, a deeper version of the Prime Number Theorem that applies to arithmetic progressions. The journey begins with fundamental tools such as Dirichlet series, Euler products, and the Riemann zeta function, focusing on their convergence and analytic continuation. Along the way, key estimates involving Chebyshev functions and binomial coefficients are developed to support later asymptotic analysis. A major part of the study centers on Dirichlet L-functions, especially their behavior on the line ℜ(s) = 1, which plays a crucial role in proving results about the distribution of primes in arithmetic sequences. To move from these analytic foundations to actual asymptotic formulas, the Wiener–Ikehara Tauberian theorem is applied. The thesis also establishes zero-free regions for ζ(s) and L(s, χ), using tools like Jensen’s inequality and the Borel–Carath´eodory lemma. The main result is a complete proof of the Siegel–Walfisz theorem. This gives strong, uniform estimates for how primes are distributed in arithmetic progressions, even when the modulus is relatively large. Specifically, for any fixed A > 0 and modulus q ≤ (log x)A, it shows: ψ(x; q, a) = x φ(q) + OA � x exp � −c p log x �� , where (a, q) = 1. The proof carefully handles the tricky issue of exceptional zeros, relying on Siegel’s theorem to keep their effects under control. Overall, the thesis brings together a range of analytic and algebraic techniques to offer a clear and connected view of how deep results in number theory build on one another. It’s aimed at helping readers see the big picture while appreciating the precision of the tools involved. Keywords: Prime Number Theorem, Siegel–Walfisz theorem, Dirichlet L-functions, Tauberian theorems, exceptional zeros, Chebyshev estimates. 5

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