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Notes on analytic number theory
Kiran S. Kedlaya
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Front Matter
Colophon
About this document
Introduction to the course
I
Distribution of primes
1
The prime number theorem
1.1
Euler’s idea: revisiting the infinitude of primes
1.2
Riemann’s zeta function
1.3
Towards the prime number theorem
1.4
The Tauberian argument
1.5
Historical aside: the Erdős–Selberg method
1.6
Exercises
2
Dirichlet series and arithmetic functions
2.1
Dirichlet series
2.2
Multiplicative functions and Euler products
2.3
Examples of multiplicative functions
2.4
Exercises
3
Dirichlet characters and
\(L\)
-functions
3.1
Dirichlet characters
3.2
\(L\)
-series
3.3
Nonvanishing of
\(L\)
-functions on
\(\Real(s) = 1\)
3.4
Nonvanishing for
\(L\)
-functions at
\(s=1\)
3.5
Historical aside: Dirichlet’s class number formula
3.6
Exercises
4
Primes in arithmetic progressions
4.1
Dirichlet’s theorem
4.2
Asymptotic density and Dirichlet density
4.3
\(L\)
-functions and discrete Fourier analysis
4.4
The prime number theorem in arithmetic progressions
4.5
Exercises
II
Error estimates from
\(L\)
-functions
5
The functional equation for the Riemann zeta function
5.1
The functional equation for
\(\zeta\)
5.2
The
\(\theta\)
function and the Fourier transform
5.3
Asides
5.4
Exercises
6
Functional equations for Dirichlet
\(L\)
-functions
6.1
Even characters
6.2
Odd characters
6.3
Exercises
7
Error bounds in the prime number theorem
7.1
Zeta zeroes and prime numbers
7.2
How to use von Mangoldt’s formula
7.3
The Riemann Hypothesis
7.4
Exercises
8
More on the zeroes of zeta
8.1
Order of an entire function
8.2
Product representations of entire functions
8.3
A zero-free region for
\(\zeta\)
8.4
Exercises
9
von Mangoldt’s formula
9.1
Truncating a Dirichlet series
9.2
The effect of shifting contours
9.3
Truncating the vertical integral
9.4
Bounds on
\(\zeta'/\zeta\)
9.5
Final assembly
9.6
Exercises
10
Error bounds for primes in arithmetic progressions
10.1
von Mangoldt’s formula for
\(L\)
-functions
10.2
Uniformity in the explicit formula
10.3
The generalized Riemann hypothesis
10.4
Zero-free regions for
\(\chi\)
10.5
Controlling the exceptional zeroes
III
Introduction to sieve methods
11
Revisiting the sieve of Eratosthenes
11.1
The sieve of Eratosthenes
11.2
The principle of inclusion-exclusion
11.3
Smooth numbers
11.4
Back to Eratosthenes
11.5
Motivation: the twin prime conjecture
11.6
Exercises
12
Brun’s combinatorial sieve
12.1
Sieve setup
12.2
Brun’s combinatorial sieve
12.3
Setting some parameters
12.4
Bounding the main term
12.5
Consequences for twin almost-primes
12.6
Exercises
13
The Selberg sieve
13.1
Review of notation
13.2
The Selberg upper bound sieve
13.3
Exercises
14
Applying the Selberg sieve
14.1
Bounding sums of multiplicative functions
14.2
Bounding the main term
14.3
Bounding the error term
14.4
Exercises
IV
Large sieves
15
An additive large sieve inequality
15.1
Overview
15.2
Preparatory lemmas
15.3
An additive large sieve
15.4
Exercises
16
A multiplicative large sieve inequality
16.1
A specialization of the additive large sieve
16.2
The Bombieri–Davenport inequality
16.3
An application of the large sieve
16.4
Exercises
17
The Bombieri–Vinogradov theorem: statement
17.1
Statement of the theorem
17.2
Exercises
18
The Bombieri–Vinogradov theorem: proof
18.1
Bounding character sums
18.2
Proof of the theorem
18.3
The Barban–Davenport–Halberstam theorem
18.4
Exercises
V
Gaps between primes
19
Prime
\(k\)
-tuples
19.1
The Hardy–Littlewood
\(k\)
-tuples conjecture
19.2
\(k\)
-tuples and prime gaps
19.3
Exercises
20
Bounded gaps between primes: outline
20.1
The goal: a weakening of the
\(k\)
-tuples conjecture
20.2
Algebraic setup
20.3
Revisiting the Selberg sieve
20.4
Identification of error terms
20.5
Optimizing the objective function
20.6
Exercises
21
Bounded gaps between primes: proofs
21.1
Review of the setup
21.2
Computing the main terms
21.3
Estimating the error term
21.4
Application of Bombieri–Vinogradov
21.5
Optimizing the objective function
VI
Additional topics
22
Artin
\(L\)
-functions and the Chebotaryov density theorem
22.1
Frobenius elements of Galois groups
22.2
Linear representations and
\(L\)
-functions
22.3
Artin’s conjecture
22.4
Induced representations
22.5
Chebotaryov’s density theorem
22.6
Exercises
23
Elliptic curves and their
\(L\)
-functions
23.1
Elliptic curves and their
\(L\)
-functions
24
The Sato–Tate distribution
24.1
Equidistribution on compact groups
24.2
Topological groups
24.3
\(L\)
-functions and equidistribution
24.4
The Sato–Tate conjecture
24.5
Equidistribution and Sato–Tate
24.6
Exercises
Back Matter
Bibliography
Colophon
Colophon
Website
https://kskedlaya.org
©2015–2025 Kiran S. Kedlaya
This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit
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🔗
