1
Introduction
2
Basic notation
▶
2.1
Asymptotic (or “cheap nonstandard”) notation
3
Basic Fourier estimates
4
Exponential sum growth exponents
▶
4.1
Phase functions
4.2
Exponential sum exponent
4.3
Known bounds on \(\beta \)
5
Exponent pairs
▶
5.1
Known exponent pairs
6
Growth exponents for the Riemann zeta function
▶
6.1
Connection with exponent pairs and dual exponent pairs
6.2
Known bounds on \(\mu \)
6.3
Connection to the Riemann hypothesis
7
Large value estimates
▶
7.1
Known upper bounds on \(\mathrm{LV}(\sigma ,\tau )\)
8
Large value theorems for zeta partial sums
9
Moment growth for the zeta function
▶
9.1
Relationship to zeta large value estimates
9.2
Known moment growth bounds
9.3
Large values of \(\zeta \) moments
10
Large value additive energy
▶
10.1
Additive energy
10.2
Large value additive energy region
10.3
Known relations for the large value energy region
11
Zero density theorems
▶
11.1
Known zero density bounds
11.2
Estimates for \(\sigma \) very close to \(1/2\) or \(1\)
11.3
A heuristic for zero density estimates
11.4
Explicit results
12
Zero density energy theorems
▶
12.1
Known additive energy bounds
13
Zero free regions
▶
13.1
Relation to growth rates of zeta
14
Distribution of primes: long ranges
▶
14.1
Error bounds for prime counting functions
14.2
Relation to zero free region of zeta
14.3
Omega results
15
Distribution of primes: short ranges
▶
15.1
Extremal values of prime gaps
16
The generalized Dirichlet divisor problem
▶
16.1
Known pointwise bounds on divisor sum exponents
17
The number of Pythagorean triples
18
The de Bruijn–Newman constant
19
Brun-Titchmarsh type theorems
▶
19.1
Upper bounds
19.2
Lower bounds
20
Waring and Goldbach type problems, and Schnirelman’s constant
▶
20.1
Waring Problem
▶
20.1.1
Known values of g(k)
20.1.2
Known values of \(G(k)\)
20.1.3
General bounds for \(G(k)\)
20.1.4
Bounds for special cases for \(G(k)\)
20.1.5
Generalized Waring problem and connections to the Generalized Riemann Hypothesis
20.2
Goldbach-Style Problems
▶
20.2.1
When \(k=2\)
20.2.2
When \(k=4,5\)
20.2.3
When \(k\ge 7\)
20.3
Schnirelmann Density
▶
20.3.1
Existence of Additive Basis
20.3.2
Essential Components
21
The Gauss circle problem and its generalizations
▶
21.1
Known upper and lower bounds
22
Bibliography
Dependency graph
Analytic Number Theory Exponent Database
Terence Tao, Timothy Trudgian, Andrew Yang
1
Introduction
2
Basic notation
2.1
Asymptotic (or “cheap nonstandard”) notation
3
Basic Fourier estimates
4
Exponential sum growth exponents
4.1
Phase functions
4.2
Exponential sum exponent
4.3
Known bounds on \(\beta \)
5
Exponent pairs
5.1
Known exponent pairs
6
Growth exponents for the Riemann zeta function
6.1
Connection with exponent pairs and dual exponent pairs
6.2
Known bounds on \(\mu \)
6.3
Connection to the Riemann hypothesis
7
Large value estimates
7.1
Known upper bounds on \(\mathrm{LV}(\sigma ,\tau )\)
8
Large value theorems for zeta partial sums
9
Moment growth for the zeta function
9.1
Relationship to zeta large value estimates
9.2
Known moment growth bounds
9.3
Large values of \(\zeta \) moments
10
Large value additive energy
10.1
Additive energy
10.2
Large value additive energy region
10.3
Known relations for the large value energy region
11
Zero density theorems
11.1
Known zero density bounds
11.2
Estimates for \(\sigma \) very close to \(1/2\) or \(1\)
11.3
A heuristic for zero density estimates
11.4
Explicit results
12
Zero density energy theorems
12.1
Known additive energy bounds
13
Zero free regions
13.1
Relation to growth rates of zeta
14
Distribution of primes: long ranges
14.1
Error bounds for prime counting functions
14.2
Relation to zero free region of zeta
14.3
Omega results
15
Distribution of primes: short ranges
15.1
Extremal values of prime gaps
16
The generalized Dirichlet divisor problem
16.1
Known pointwise bounds on divisor sum exponents
17
The number of Pythagorean triples
18
The de Bruijn–Newman constant
19
Brun-Titchmarsh type theorems
19.1
Upper bounds
19.2
Lower bounds
20
Waring and Goldbach type problems, and Schnirelman’s constant
20.1
Waring Problem
20.1.1
Known values of g(k)
20.1.2
Known values of \(G(k)\)
20.1.3
General bounds for \(G(k)\)
20.1.4
Bounds for special cases for \(G(k)\)
20.1.5
Generalized Waring problem and connections to the Generalized Riemann Hypothesis
20.2
Goldbach-Style Problems
20.2.1
When \(k=2\)
20.2.2
When \(k=4,5\)
20.2.3
When \(k\ge 7\)
20.3
Schnirelmann Density
20.3.1
Existence of Additive Basis
20.3.2
Essential Components
21
The Gauss circle problem and its generalizations
21.1
Known upper and lower bounds
22
Bibliography