1
The project
2
First approach: Wiener-Ikehara Tauberian theorem
▶
2.1
A Fourier-analytic proof of the Wiener-Ikehara theorem
2.2
Weak PNT
2.3
Removing the Chebyshev hypothesis
2.4
The prime number theorem in arithmetic progressions
2.5
The Chebotarev density theorem: the case of cyclotomic extensions
2.6
The Chebotarev density theorem: the case of abelian extensions
2.7
The Chebotarev density theorem: the general case
3
Second approach
▶
3.1
Residue calculus on rectangles
3.2
Perron Formula
3.3
Mellin transforms
3.4
Zeta Bounds
3.5
Proof of Medium PNT
3.6
MediumPNT
4
Third Approach
▶
4.1
Hadamard factorization
4.2
Hoffstein-Lockhart
4.3
Strong PNT
5
Elementary Corollaries
▶
5.1
Consequences of the PNT in arithmetic progressions
5.2
Consequences of the Chebotarev density theorem
6
Explicit estimates
7
Zeta function estimates
▶
7.1
Definitions
7.2
The estimates of Kadiri, Lumley, and Ng
7.3
The zeta function bounds of Rosser and Schoenfeld
7.4
Approximating the Riemann zeta function
8
Primary explicit estimates
▶
8.1
Definitions
8.2
A Lemma involving the Möbius Function
8.3
The estimates of Fiori, Kadiri, and Swidinsky
8.4
Numerical content of BKLNW Appendix A
8.5
Appendix A of BKLNW
8.6
Chirre-Helfgott’s estimates for sums of nonnegative arithmetic functions
8.7
Summary of results
9
Secondary explicit estimates
▶
9.1
Definitions
9.2
Chebyshev’s estimates
9.3
An inequality of Costa-Pereira
9.4
The prime number bounds of Rosser and Schoenfeld
9.5
The estimates of Buthe
9.6
Numerical content of BKLNW
9.7
Tools from BKLNW
9.8
The implications of FKS2
9.9
Dusart’s explicit estimates for primes
9.10
Summary of results
10
Tertiary explicit estimates
▶
10.1
The least common multiple sequence is not highly abundant for large \(n\)
10.2
Erdos problem 392
11
Iwaniec-Kowalski
▶
11.1
Blueprint for Iwaniec-Kowalski Chapter 1
12
Bibliography
Dependency graph
Prime Number Theorem And ...
1
The project
2
First approach: Wiener-Ikehara Tauberian theorem
2.1
A Fourier-analytic proof of the Wiener-Ikehara theorem
2.2
Weak PNT
2.3
Removing the Chebyshev hypothesis
2.4
The prime number theorem in arithmetic progressions
2.5
The Chebotarev density theorem: the case of cyclotomic extensions
2.6
The Chebotarev density theorem: the case of abelian extensions
2.7
The Chebotarev density theorem: the general case
3
Second approach
3.1
Residue calculus on rectangles
3.2
Perron Formula
3.3
Mellin transforms
3.4
Zeta Bounds
3.5
Proof of Medium PNT
3.6
MediumPNT
4
Third Approach
4.1
Hadamard factorization
4.2
Hoffstein-Lockhart
4.3
Strong PNT
5
Elementary Corollaries
5.1
Consequences of the PNT in arithmetic progressions
5.2
Consequences of the Chebotarev density theorem
6
Explicit estimates
7
Zeta function estimates
7.1
Definitions
7.2
The estimates of Kadiri, Lumley, and Ng
7.3
The zeta function bounds of Rosser and Schoenfeld
7.4
Approximating the Riemann zeta function
8
Primary explicit estimates
8.1
Definitions
8.2
A Lemma involving the Möbius Function
8.3
The estimates of Fiori, Kadiri, and Swidinsky
8.4
Numerical content of BKLNW Appendix A
8.5
Appendix A of BKLNW
8.6
Chirre-Helfgott’s estimates for sums of nonnegative arithmetic functions
8.7
Summary of results
9
Secondary explicit estimates
9.1
Definitions
9.2
Chebyshev’s estimates
9.3
An inequality of Costa-Pereira
9.4
The prime number bounds of Rosser and Schoenfeld
9.5
The estimates of Buthe
9.6
Numerical content of BKLNW
9.7
Tools from BKLNW
9.8
The implications of FKS2
9.9
Dusart’s explicit estimates for primes
9.10
Summary of results
10
Tertiary explicit estimates
10.1
The least common multiple sequence is not highly abundant for large \(n\)
10.2
Erdos problem 392
11
Iwaniec-Kowalski
11.1
Blueprint for Iwaniec-Kowalski Chapter 1
12
Bibliography