# 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…

## 2,748 Citations

Twisted second moments and explicit formulae of the Riemann zeta-function

- Mathematics
- 2015

Several aspects connecting analytic number theory and the Riemann zeta-function are studied and expanded. These include:
1. explicit formulae relating the Mobius function to the non-trivial zeros of…

Discrete Mean Values of Dirichlet L-functions

- Mathematics
- 2015

In 1911 Landau proved an asymptotic formula for sums of the form ∑ γ≤T x ρ over the imaginary parts of the nontrivial zeros of the Riemann zeta function. The formula provided yet another deep…

Arithmetic exponent pairs for algebraic trace functions and applications

- MathematicsAlgebra & Number Theory
- 2021

We study short sums of algebraic trace functions via the $q$-analogue of van der Corput method, and develop methods of arithmetic exponent pairs that coincide with the classical case while the moduli…

Sums of singular series and primes in short intervals in algebraic number fields

- MathematicsThe Ramanujan Journal
- 2022

Gross and Smith have put forward generalizations of Hardy - Littlewood twin prime conjectures for algebraic number fields. We estimate the behavior of sums of a singular series that arises in these…

Topics in analytic number theory

- Mathematics
- 2016

In this thesis, we investigate three topics belonging to the probabilistic, classical and modern branches of analytic number theory. Our first result concerns the probabilistic distribution of…

Character sum estimates in finite fields and applications

- Mathematics, Computer Science
- 2015

This thesis presents a number of character sum estimates for sums of various types occurring in finite fields and gives applications of such estimates to problems in arithmetic combinatorics and analytic number theory.

Mean-periodicity and zeta functions

- Mathematics
- 2008

This paper establishes new bridges between number theory and modern harmonic analysis, namely between the class of complex functions, which contains zeta functions of arithmetic schemes and closed…

Some aspects of analytic number theory: parity, transcendence, and multiplicative functions

- Mathematics
- 2009

Questions on parities play a central role in analytic number theory. Properties of the partial sums of parities are intimate to both the prime number theorem and the Riemann hypothesis. This thesis…

Zeros of the Dedekind Zeta-Function

- Mathematics
- 2019

H. L. Montgomery proved a formula for sums over two sets of nontrivial zeros of the Riemann zeta-function. Assuming the Riemann Hypothesis, he used this formula and Fourier analysis to prove an…

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