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
β
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
μ
6.3
Connection to the Riemann hypothesis
7
Large value estimates
▶
7.1
Known upper bounds on
LV
(
σ
,
τ
)
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
ζ
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
σ
very close to
1
/
2
or
1
11.3
A heuristic for zero density estimates
12
Zero density energy theorems
▶
12.1
Known additive energy bounds
13
Applications to the primes
14
The generalized Dirichlet divisor problem
▶
14.1
Known pointwise bounds on divisor sum exponents
15
The number of Pythagorean triples
16
The de Bruijn–Newman constant
17
The Prime Counting Function
▶
17.1
Introduction
17.2
The Prime Number Theorem and Asymptotic Behavior
17.3
Upper Bounds for
π
(
x
)
▶
17.3.1
Chebyshev’s Upper Bound
17.4
Lower Bounds for
π
(
x
)
▶
17.4.1
A Classical Lower Bound
17.5
Sharper Upper and Lower Bounds
17.6
Alternative Approximations
17.7
Computational Aspects
17.8
Error Terms in the Prime Number Theorem
17.9
Future Research Directions
17.10
Conclusion
17.11
References
18
Zero free region for the zeta function
▶
18.1
Relation to upper bound on zeta in the critical strip
18.2
Relation to the error term in the prime number theorem
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
≥
7
20.3
Schnirelmann Density
▶
20.3.1
Existence of Additive Basis
20.3.2
Essential Components
21
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
β
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
μ
6.3
Connection to the Riemann hypothesis
7
Large value estimates
7.1
Known upper bounds on
LV
(
σ
,
τ
)
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
ζ
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
σ
very close to
1
/
2
or
1
11.3
A heuristic for zero density estimates
12
Zero density energy theorems
12.1
Known additive energy bounds
13
Applications to the primes
14
The generalized Dirichlet divisor problem
14.1
Known pointwise bounds on divisor sum exponents
15
The number of Pythagorean triples
16
The de Bruijn–Newman constant
17
The Prime Counting Function
17.1
Introduction
17.2
The Prime Number Theorem and Asymptotic Behavior
17.3
Upper Bounds for
π
(
x
)
17.3.1
Chebyshev’s Upper Bound
17.4
Lower Bounds for
π
(
x
)
17.4.1
A Classical Lower Bound
17.5
Sharper Upper and Lower Bounds
17.6
Alternative Approximations
17.7
Computational Aspects
17.8
Error Terms in the Prime Number Theorem
17.9
Future Research Directions
17.10
Conclusion
17.11
References
18
Zero free region for the zeta function
18.1
Relation to upper bound on zeta in the critical strip
18.2
Relation to the error term in the prime number theorem
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
≥
7
20.3
Schnirelmann Density
20.3.1
Existence of Additive Basis
20.3.2
Essential Components
21
Bibliography