Chapter 3. Analytic Number Theory – brain

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

Chapter 3. Analytic Number TheoryThe harmonic series is the infinite series

Infinite ProductsAn infinite product has the form

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

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

Abel SummationIn analytic number theory, one often studies sums of the form

Prime Counting FunctionOne 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,...

Chebyshev BoundsThe prime counting function

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

Logarithmic IntegralThe logarithmic integral is the function

Error TermsThe Prime Number Theorem states that

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

Twin Prime HeuristicsA pair of primes

Definition of the Zeta FunctionOne 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...

Euler Product FormulaThe defining series of the Riemann zeta function is

Analytic ContinuationThe defining series of the Riemann zeta function is

Functional EquationThe defining series of the zeta function,

Zeros of the Zeta FunctionThe zeros of the Riemann zeta function are the complex numbers $s$ satisfying

Riemann HypothesisThe Riemann zeta function has nontrivial zeros inside the critical strip

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

Connections with Prime DistributionThe Riemann zeta function was introduced through the series

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

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

Dirichlet $L$-FunctionsThe Riemann zeta function

Primes in Arithmetic ProgressionsAn arithmetic progression is a sequence of the form

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

Generalized Riemann HypothesisThe classical Riemann Hypothesis concerns the zeros of the Riemann zeta function

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

Goldbach ProblemsGoldbach-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.

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

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

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

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

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

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

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

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

Brun-Titchmarsh TheoremOne of the central problems of analytic number theory is understanding how primes distribute among residue classes.

Bombieri-Vinogradov TheoremThe Prime Number Theorem for arithmetic progressions states that for

Chen's TheoremThe Twin Prime Conjecture states that infinitely many primes satisfy

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