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Notes on analytic number theory
Kiran S. Kedlaya
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Front Matter
Colophon
Preface
1
Introduction to the course
1.1
What is analytic number theory?
1.2
Basic structure of the course
1.3
Notations and conventions
Basics
Asymptotics
Miscellaneous
I
Distribution of primes
2
The prime number theorem
2.1
Euler's idea: revisiting the infinitude of primes
2.2
Riemann's zeta function
2.3
Towards the prime number theorem
2.4
The Tauberian argument
2.5
Historical aside: the Erdős-Selberg method
2.6
Exercises
3
Dirichlet series and arithmetic functions
3.1
Dirichlet series
3.2
Multiplicative functions and Euler products
3.3
Examples of multiplicative functions
3.4
Exercises
4
Dirichlet characters and
\(L\)
-functions
4.1
Dirichlet characters
4.2
\(L\)
-series
4.3
Nonvanishing of
\(L\)
-functions on
\(\Real(s) = 1\)
4.4
Nonvanishing for
\(L\)
-functions at
\(s=1\)
4.5
Historical aside: Dirichlet's class number formula
4.6
Exercises
5
Primes in arithmetic progressions
5.1
Dirichlet's theorem
5.2
Asymptotic density and Dirichlet density
5.3
\(L\)
-functions and discrete Fourier analysis
5.4
The prime number theorem in arithmetic progressions
5.5
Exercises
II
Error estimates from
\(L\)
-functions
6
The functional equation for the Riemann zeta function
6.1
The functional equation for
\(\zeta\)
6.2
The
\(\theta\)
function and the Fourier transform
6.3
Asides
6.4
Exercises
7
Functional equations for Dirichlet
\(L\)
-functions
7.1
Even characters
7.2
Odd characters
7.3
Exercises
8
Error bounds in the prime number theorem
8.1
Zeta zeroes and prime numbers
8.2
How to use von Mangoldt's formula
8.3
The Riemann Hypothesis
8.4
Exercises
9
More on the zeroes of zeta
9.1
Order of an entire function
9.2
Product representations of entire functions
9.3
A zero-free region for
\(\zeta\)
9.4
Exercises
10
von Mangoldt's formula
10.1
Truncating a Dirichlet series
10.2
The effect of shifting contours
10.3
Truncating the vertical integral
10.4
Bounds on
\(\zeta'/\zeta\)
10.5
Final assembly
10.6
Exercises
11
Error bounds for primes in arithmetic progressions
11.1
von Mangoldt's formula for
\(L\)
-functions
11.2
Uniformity in the explicit formula
11.3
The generalized Riemann hypothesis
11.4
Zero-free regions for
\(\chi\)
11.5
Controlling the exceptional zeroes
III
Introduction to sieve methods
12
Revisiting the sieve of Eratosthenes
12.1
The sieve of Eratosthenes
12.2
The principle of inclusion-exclusion
12.3
Smooth numbers
12.4
Back to Eratosthenes
12.5
Motivation: the twin prime conjecture
12.6
Exercises
13
Brun's combinatorial sieve
13.1
Sieve setup
13.2
Brun's combinatorial sieve
13.3
Setting some parameters
13.4
Bounding the main term
13.5
Consequences for twin almost-primes
13.6
Exercises
14
The Selberg sieve
14.1
Review of notation
14.2
The Selberg upper bound sieve
14.3
Exercises
15
Applying the Selberg sieve
15.1
Bounding sums of multiplicative functions
15.2
Bounding the main term
15.3
Bounding the error term
15.4
Exercises
IV
Large sieves
16
An additive large sieve inequality
16.1
Overview
16.2
Preparatory lemmas
16.3
An additive large sieve
16.4
Exercises
17
A multiplicative large sieve inequality
17.1
A specialization of the additive large sieve
17.2
The Bombieri–Davenport inequality
17.3
An application of the large sieve
17.4
Exercises
18
The Bombieri–Vinogradov theorem: statement
18.1
Statement of the theorem
18.2
Exercises
19
The Bombieri–Vinogradov theorem: proof
19.1
Bounding character sums
19.2
Proof of the theorem
19.3
The Barban-Davenport-Halberstam theorem
19.4
Exercises
V
Gaps between primes
20
Prime
\(k\)
-tuples
20.1
The Hardy–Littlewood
\(k\)
-tuples conjecture
20.2
\(k\)
-tuples and prime gaps
20.3
Exercises
21
Bounded gaps between primes: outline
21.1
The goal: a weakening of the
\(k\)
-tuples conjecture
21.2
Algebraic setup
21.3
Revisiting the Selberg sieve
21.4
Identification of error terms
21.5
Optimizing the objective function
21.6
Exercises
22
Bounded gaps between primes: proofs
22.1
Review of the setup
22.2
Computing the main terms
22.3
Estimating the error term
22.4
Application of Bombieri–Vinogradov
22.5
Optimizing the objective function
VI
Additional topics
23
Artin
\(L\)
-functions and the Chebotaryov density theorem
23.1
Frobenius elements of Galois groups
23.2
Linear representations and
\(L\)
-functions
23.3
Artin's conjecture
23.4
Induced representations
23.5
Chebotaryov's density theorem
23.6
Exercises
24
Elliptic curves and their
\(L\)
-functions
24.1
Elliptic curves and their
\(L\)
-functions
25
The Sato–Tate distribution
25.1
Equidistribution on compact groups
25.2
Topological groups
25.3
\(L\)
-functions and equidistribution
25.4
The Sato–Tate conjecture
25.5
Equidistribution and Sato–Tate
25.6
Exercises
Back Matter
Bibliography
Colophon
Colophon
https://kskedlaya.org/ant
©2015–2023 Kiran S. Kedlaya
