Chapter 3. Analytic Number Theory

The harmonic series is the infinite series

41 items

Section	Title
1	Chapter 3. Analytic Number Theory
2	Infinite Products
3	Euler Products
4	Convergence Methods
5	Abel Summation
6	Prime Counting Function
7	Chebyshev Bounds
8	Prime Number Theorem
9	Logarithmic Integral
10	Error Terms
11	Short Intervals
12	Prime Gaps
13	Twin Prime Heuristics
14	Definition of the Zeta Function
15	Euler Product Formula
16	Analytic Continuation
17	Functional Equation
18	Zeros of the Zeta Function
19	Riemann Hypothesis
20	Explicit Formulae
21	Connections with Prime Distribution
22	Dirichlet Characters
23	Orthogonality Relations
24	Dirichlet $L$-Functions
25	Primes in Arithmetic Progressions
26	Nonvanishing Results
27	Generalized Riemann Hypothesis
28	Sumsets
29	Goldbach Problems
30	Waring's Problem
31	Circle Method
32	Exponential Sums
33	Additive Bases
34	Schnirelmann Density
35	Brun Sieve
36	Selberg Sieve
37	Large Sieve
38	Brun-Titchmarsh Theorem
39	Bombieri-Vinogradov Theorem
40	Chen's Theorem
41	Parity Problem

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Chapter 3. Analytic Number Theory

The harmonic series is the infinite series

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Infinite Products

An infinite product has the form

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Euler Products

Euler products arise when an infinite series has coefficients controlled by multiplication. The simplest and most important example is the zeta series

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Convergence Methods

Analytic number theory studies infinite sums, products, and integrals. Before such expressions can be manipulated safely, one must understand the meaning of convergence.

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Abel Summation

In analytic number theory, one often studies sums of the form

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Prime Counting Function

One of the oldest questions in number theory asks how prime numbers are distributed among the positive integers. Since primes become less frequent as numbers grow larger,...

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Chebyshev Bounds

The prime counting function

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Prime Number Theorem

The Prime Number Theorem describes the asymptotic distribution of prime numbers. It states that

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Logarithmic Integral

The logarithmic integral is the function

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Error Terms

The Prime Number Theorem states that

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Short Intervals

The Prime Number Theorem describes the average distribution of primes up to a large number $x$:

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Prime Gaps

Let

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Twin Prime Heuristics

A pair of primes

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Definition of the Zeta Function

One of the central objects of analytic number theory is the Riemann zeta function. It connects infinite series, prime numbers, complex analysis, and arithmetic structure into...

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Euler Product Formula

The defining series of the Riemann zeta function is

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Analytic Continuation

The defining series of the Riemann zeta function is

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Functional Equation

The defining series of the zeta function,

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Zeros of the Zeta Function

The zeros of the Riemann zeta function are the complex numbers $s$ satisfying

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Riemann Hypothesis

The Riemann zeta function has nontrivial zeros inside the critical strip

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Explicit Formulae

One of the deepest ideas in analytic number theory is that the zeros of the zeta function determine the distribution of prime numbers.

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Connections with Prime Distribution

The Riemann zeta function was introduced through the series

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Dirichlet Characters

The Riemann zeta function studies prime numbers globally, without distinguishing congruence classes. However, many arithmetic questions concern primes satisfying conditions such as

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Orthogonality Relations

Dirichlet characters behave analogously to exponential functions in Fourier analysis. Just as complex exponentials separate frequencies, characters separate residue classes...

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Dirichlet $L$-Functions

The Riemann zeta function

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Primes in Arithmetic Progressions

An arithmetic progression is a sequence of the form

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Nonvanishing Results

A central theme in analytic number theory is determining when an $L$-function is nonzero at a particular point.

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Generalized Riemann Hypothesis

The classical Riemann Hypothesis concerns the zeros of the Riemann zeta function

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Sumsets

Additive number theory studies arithmetic structure through addition of integers and subsets of integers.

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Goldbach Problems

Goldbach-type problems ask whether integers can be represented as sums of primes. They are among the oldest and most famous problems in additive number theory.

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Waring's Problem

Waring's problem asks whether every sufficiently large positive integer can be written as a sum of a bounded number of fixed powers.

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Circle Method

Many problems in additive number theory ask whether an integer can be represented in the form

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Exponential Sums

Exponential sums are among the central tools of analytic number theory.

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Additive Bases

A central question in additive number theory asks whether every integer can be represented as a sum of elements from a fixed set.

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Schnirelmann Density

In additive number theory, ordinary asymptotic density is often too weak to control additive behavior.

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Brun Sieve

Sieve methods are techniques for counting integers that remain after removing residue classes modulo primes.

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Selberg Sieve

Brun's sieve introduced the idea of estimating sifted sets through truncated inclusion-exclusion. However, Brun's method often produced bounds that were technically difficult...

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Large Sieve

Classical sieve methods estimate how many integers survive congruence restrictions. The large sieve approaches these problems from a different direction.

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Brun-Titchmarsh Theorem

One of the central problems of analytic number theory is understanding how primes distribute among residue classes.

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Bombieri-Vinogradov Theorem

The Prime Number Theorem for arithmetic progressions states that for

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Chen's Theorem

The Twin Prime Conjecture states that infinitely many primes satisfy

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Parity Problem

Sieve methods are extremely effective for estimating how many integers avoid small prime factors. They have produced major results about: