Statement: Prove the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions, which asserts that all non-trivial zeros of Dirichlet L-functions lie on the critical line with real part 1/2. The solution should cover the background on Dirichlet characters, L-functions, and the analytic continuation of these functions. Discuss the role of GRH in number theory, particularly in the distribution of primes in arithmetic progressions. Required Research: 1. "Generalized Riemann Hypothesis and Dirichlet L-functions" [https://www.math.ucla.edu/~tao/246a.1.12s/notes/246A/notes15.pdf] 2. "Applications of the Generalized Riemann Hypothesis in Number Theory" [https://www.jstor.org/stable/43271160] 3. "Dirichlet L-functions and Their Critical Zeros" [https://www.ams.org/journals/proc/1990-107- 03/S0002-9939-1990-1032089-3]

Statement: Prove the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions, which asserts that all non-trivial zeros of Dirichlet L-functions lie on the critical line with real part 1/2. The solution should cover the background on Dirichlet characters, L-functions, and the analytic continuation of these functions. Discuss the role of GRH in number theory, particularly in the distribution of primes in arithmetic progressions. Required Research: 1. "Generalized Riemann Hypothesis and Dirichlet L-functions" [https://www.math.ucla.edu/~tao/246a.1.12s/notes/246A/notes15.pdf] 2. "Applications of the Generalized Riemann Hypothesis in Number Theory" [https://www.jstor.org/stable/43271160] 3. "Dirichlet L-functions and Their Critical Zeros" [https://www.ams.org/journals/proc/1990-107- 03/S0002-9939-1990-1032089-3]

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section: Chapter Questions

Problem 22RE

See similar textbooks