1
The project
2
Basic definitions and notation
3
First approach: Wiener-Ikehara Tauberian theorem
▶
3.1
A Fourier-analytic proof of the Wiener-Ikehara theorem
3.2
Weak PNT
3.3
Removing the Chebyshev hypothesis
3.4
The prime number theorem in arithmetic progressions
3.5
The Chebotarev density theorem: the case of cyclotomic extensions
3.6
The Chebotarev density theorem: the case of abelian extensions
3.7
The Chebotarev density theorem: the general case
4
Second approach
▶
4.1
Residue calculus on rectangles
4.2
Perron Formula
4.3
Mellin transforms
4.4
Zeta Bounds
4.5
Proof of Medium PNT
4.6
MediumPNT
5
Third Approach
▶
5.1
Hadamard factorization
5.2
Hoffstein-Lockhart
5.3
Strong PNT
6
Elementary Corollaries
▶
6.1
Consequences of the PNT in arithmetic progressions
6.2
Consequences of the Chebotarev density theorem
7
Explicit estimates
8
Zeta function estimates
▶
8.1
Definitions
8.2
The estimates of Kadiri, Lumley, and Ng
8.3
The zeta function bounds of Rosser and Schoenfeld
8.4
Approximating the Riemann zeta function
9
Primary explicit estimates
▶
9.1
Definitions
9.2
A Lemma involving the Möbius Function
9.3
The estimates of Fiori, Kadiri, and Swidinsky
9.4
Numerical content of BKLNW Appendix A
9.5
Appendix A of BKLNW
9.6
Chirre-Helfgott’s estimates for sums of nonnegative arithmetic functions
9.7
Summary of results
10
Secondary explicit estimates
▶
10.1
Definitions
10.2
Chebyshev’s estimates
10.3
An inequality of Costa-Pereira
10.4
Converting prime number theorems to prime in short interval theorems
10.5
The prime number bounds of Rosser and Schoenfeld
10.6
The estimates of Buthe
10.7
Numerical content of BKLNW
10.8
Tools from BKLNW
10.9
The implications of FKS2
10.10
Numerical content of eSHP
10.11
Prime gap data from eSHP
10.12
Dusart’s explicit estimates for primes
10.13
Summary of results
11
Tertiary explicit estimates
▶
11.1
The least common multiple sequence is not highly abundant for large \(n\)
11.2
Erdos problem 392
11.3
Numerical verification of Goldbach
11.4
Ramanujan’s inequality
12
Iwaniec-Kowalski
▶
12.1
Blueprint for Iwaniec-Kowalski Chapter 1
13
Bibliography
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