Multiplicative Number Theory I: Classical TheoryPrime numbers are the multiplicative building blocks of natural numbers. Understanding their overall influence and especially their distribution gives rise to central questions in mathematics and physics. In particular their finer distribution is closely connected with the Riemann hypothesis, the most important unsolved problem in the mathematical world. Assuming only subjects covered in a standard degree in mathematics, the authors comprehensively cover all the topics met in first courses on multiplicative number theory and the distribution of prime numbers. They bring their extensive and distinguished research expertise to bear in preparing the student for intelligent reading of the more advanced research literature. This 2006 text, which is based on courses taught successfully over many years at Michigan, Imperial College and Pennsylvania State, is enriched by comprehensive historical notes and references as well as over 500 exercises. |
Contents
The elementary theory of arithmetic functions | 35 |
Principles and first examples of sieve methods | 76 |
I | 108 |
II | 137 |
The Prime Number Theorem | 168 |
Applications of the Prime Number Theorem | 199 |
Further discussion of the Prime Number Theorem | 244 |
Primitive characters and Gauss sums | 282 |
Conditional estimates | 419 |
Zeros | 452 |
Oscillations of error terms | 463 |
APPENDICES | 486 |
Notes | 492 |
Notes | 513 |
The gamma function | 520 |
Topics in harmonic analysis | 535 |
Analytic properties of the zeta function and Lfunctions | 326 |
II | 358 |
Explicit formulæ | 397 |
Other editions - View all
Multiplicative Number Theory I: Classical Theory Hugh L. Montgomery,Robert C. Vaughan No preview available - 2007 |
Multiplicative Number Theory I: Classical Theory Hugh L. Montgomery,Robert C. Vaughan No preview available - 2012 |
Common terms and phrases
a₁ absolutely convergent Acta Arith Akad Amer analytic arithmetic function Assume RH asymptotic B₁ Brun-Titchmarsh theorem c₁ character mod coefficients Corollary Deduce denote the number derive Dirichlet characters Dirichlet series Erdős error term estimate Euler Euler product exceptional zero Exercise follows formula Fourier half-plane Hence identity inequality interval L-functions Landau left-hand side Legendre symbol Lemma li(x Littlewood log qt log x)2 logx London Math main term Mellin transform mod q modulo q multiplicative non-negative number of primes number of zeros O(log obtain Perron's formula polynomial positive integer power series Prime Number Theorem primitive character modulo primitive quadratic character Proc proof of Theorem quadratic character real number Riemann Riemann Hypothesis Riemann-Stieltjes integral right-hand side Show sieve square-free suffices Suppose taking Tauberian Thales Verlag theory uniformly x(log zeta function
Popular passages
Page 519 - Some formulas for the Riemann zeta function at odd integer argument resulting from Fourier expansions of the Epstein zeta function, Acta Arith.