Analytic Number Theory

@inproceedings{Iwaniec2004AnalyticNT,
  title={Analytic Number Theory},
  author={Henryk Iwaniec and Emmanuel Kowalski},
  year={2004}
}
Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The… 
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References

SHOWING 1-10 OF 387 REFERENCES
Introduction to the arithmetic theory of automorphic functions
* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary
Topics in classical automorphic forms
Introduction The classical modular forms Automorphic forms in general The Eisenstein and the Poincare series Kloosterman sums Bounds for the Fourier coefficients of cusp forms Hecke operators
Gauss Sums, Kloosterman Sums, And Monodromy Groups
The study of exponential sums over finite fields, begun by Gauss nearly two centuries ago, has been completely transformed in recent years by advances in algebraic geometry, culminating in Deligne's
Topics in Multiplicative Number Theory
Three basic principles.- The large sieve.- Arithmetic formulations of the large sieve.- A weighted sieve and its application.- A lower bound of Roth.- Classical mean value theorems.- New mean value
The Theory of the Riemann Zeta-Function
The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects
The Riemann zeta function
DORIN GHISA Riemann Zeta function is one of the most studied transcendental functions, having in view its many applications in number theory,algebra, complex analysis, statistics, as well as in
Exponential sums over finite fields and differential equations over the complex numbers: Some interactions
A fundamental question (perhaps the fundamental question) we can ask about such an ƒ is: Does ƒ = 0 have a solution in integers? Clearly, if there exists an integer solution, then for any prime p
Algebraic Number Theory
This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary
Character sums and small eigenvalues for Г0(p)
  • H. Iwaniec
  • Mathematics
    Glasgow Mathematical Journal
  • 1985
Let Δ denote the Laplace operator acting on the space L2(Г/H) of automorphic functions with respect to a congruence group Г, square integrable over the fundamental domain F=Г/H. It is known that Δ
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