A course in analytic number theory / Marius Overholt.

This book is an introduction to analytic number theory suitable for beginning graduate students. It covers everything one expects in a first course in this field, such as growth of arithmetic functions, existence of primes in arithmetic progressions, and the Prime Number Theorem. But it also covers...

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Bibliographic Details
Main Author: Overholt, Marius, 1957- (Author)
Language:English
Published: Providence, Rhode Island : American Mathematical Society, [2014]
Series:Graduate studies in mathematics ; v. 160.
Subjects:
Number theory.
Arithmetic functions.
Physical Description:xviii, 371 pages : illustrations ; 26 cm.
Format: Book
Contents:
  • Introduction
  • 1: Arithmetic functions
  • The method of Chebyshev
  • Bertrand's postulate
  • Simple estimation techniques
  • The Mertens estimates
  • Sums over divisors
  • the hyperbola method
  • Notes
  • 2: Topics on arithmetic functions
  • The neighborhood method
  • The normal order method
  • The Mertens functions
  • Notes
  • 3: Characters and Euler products
  • The Euler product formula
  • Convergence of Dirichlet series
  • Harmonics
  • Group representations
  • Fourier analysis on finite groups
  • Primes in arithmetic porgressions
  • Gauss sums and primitive characters
  • Ther character group
  • Notes
  • 4: The circle method
  • Diophantine equations
  • The major arcs
  • The singular series
  • Weyl sums
  • An asymptotic estimate
  • Notes
  • 5: The method of contour integrals
  • The Perron formula
  • Bounds for Dirichlet L-functions
  • Notes
  • 6: The Prime Number Theorem
  • A zero-free region
  • A proof of o fht PNT
  • Notes
  • 7: The Siegel-Walfisz Theorem
  • Zero-free regions for L-functions
  • An idea of Landau
  • The theorem of Siegel
  • Teh Borel-Caratheodory lemme
  • The PNT for arithmetic progressions
  • Notes
  • 8: Mainly Analysis
  • The Poisson summatino formula
  • Theta functions
  • The gamma function
  • The functional equation of (s)
  • The functional equation of L(s,x)
  • The Hadamard factorization theorem
  • The Phragmen-Lindelof principle
  • Notes
  • 9: Euler Products and number fields
  • The Dedekind zeta function
  • The analytic class number formula
  • Class numbers of quadratic fields
  • A discriminant bound
  • The Prime Ideal Theorem - "A proof of the Ikehara theorem
  • Induced represntaitons
  • Artin L-functions
  • Notes
  • 10: Explicit Formulas
  • The von Mangoldt formula
  • The primes and RH
  • The Guinand-Weil formula
  • Notes
  • 11: Supplementary Exercises
  • Exercises
  • Solutions.