# An introduction to analytic number theory, fall 2014

### What is going on?

The lectures have ended. Thanks a lot for all participants!

There is still an instruction class on Friday 5.12, and the last exercise class is on Friday 12.12.

**Exercise sets**

### Some material of past lectures

### Lecturer

### Scope

10 sp.

### Type

Advanced studies

### Prerequisites

Basic knowledge of elementary number theory and algebra. Complex analysis I, Complex analysis II (only selected parts of this course, which can be studied by one self during the lectures if needed), basic measure and in integration theory.

### Lectures

Weeks 37-42 and 45-49, Tuesday 10-12 and Wednesday 9-12 in room C123. Some extra lecture times or changes in the times will be agreed on together during the course.

### Exams

The course will be passed by returning in a written form an appropriate number of given exercises (or by an exam/oral test if separately agreed with the lecturer). If X stands for the maximum number of points one may get by returning exercises, to pass the course one needs X/2 points. There are no different grades, just passed on non-passed.

The course proceeds somewhat rapidly to advanced things. However, to pass the course it is not required to master all the material, which allows us to touch many fascinating parts of the basic theory.

### Bibliography

A good and very readable introduction to analytic number theory is Tom Apostol: *Introduction to Analytic Number Theory*, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9

### Registration

Did you forget to register? What to do?

### Exercise and instruction classes

The excercise classes are as follows:

Class 1 on Friday 10.10.

Class 2 on Friday 31.10.

Class 3 on Friday 21.11.

Class 4 on Friday 12.12.

In (almost all) of the rest of Fridays one has instruction classes. They are held on Fridays 19.9., 26.9., 17.10., 7.11., 14.11, 28.11 and 5.12.

The idea of a instruction class is that Jääsaari is present and can discuss the exercise/course material.

Group | Day | Time | Place | Teacher |
---|---|---|---|---|

1. | Friday | 12-14 | C123 | Jesse Jääsaari |

### Logbook

Tuesday 9.9: Arithmetic functions, examples, basic properties and their inverses. Möbius inversion formula.

Wednesday 10.9: Multiplicative functions, basic properties and examples. Sum(matory) functions.. Laundau's notation. Divisor sum theorem with Dirichlet's rmainder term.

Tuesday 16.9: Heuristic proof for density of squarefree numbers. Rigorous proof with remainder term. Fourier series (beginning).

Wedensday 17.9: Approximation by Fejer-kernels, Parseval formula, convergence of absolutely summabe Fourier series, convergence for F series of piecewise differentiable and continuous functions.

Tuesday 23.9: Convergence of F series of piecewise continuous and differentiable functions, Poisson sum formula, Poisson sum formula for finite sums.

Wednesday 24.9: Evaluation of Gauss sums. Their relation to Legendre symbols. Analytic proof of Gauss' quadratic reciprocity theorem. Transformation formula of Jacobi theta function.

Tuesday 30.9: Gamma function, analytic continuation. Riemann's functional equation for the zeta function.

Wednesday 1.10: Euler summation formula. Applications: Strirlings formula. Stirlings formula for complex exponents.

Friday 3.10: Abel summation formula. Analyticity domain of convergen Dirichlet series. Abscissas of convercence and of absolute convergence.

Tuesday 7.10: Examples of Dirichlet series. The Ingham-Newman Tauberian theorem.

Wednesday 8.10: Uniqueness of coefficients of Dirichlet series, consequences, growth of inverses of arithmetic functions, multiplication of Dirichlet series, Euler products.

Monday 13.10: Perron formulas.

Tuesday 14.10: Perron formulas (continued).

Wednesday 15.10: Von Mangold and Chebyshev functions. Equivalent formulations of the prime number theorem (Landau's theorem).

Monday 3.11: Non-vanishing of the Riemann zeta-function on the right boundary of the critical strip. Proof of the prime number theorem. Outline of the strategy for Dirichlet's theorem on primes in arithmetic sequences.

Tuesday 4.11 Characters of a finite Abelian group. Explicite representations. Orthogonality relations. Dirichlet characters.

Wednesday 5.11 More on Dirichlet characters. Proof of Dirichlets theorem on primes in arithmetic sequences.

Monday 10.11:Auxiliary material: Hadamards product expansion for functions of finite order. Structure of finite Abelian groups.

Tuesday 11.11 auxiliary material from Monday continued. Quadratic excess

Wednesday 12.11 Proof of positivity of quadratic excess. Discussion of Riemann's memoir. Basic information of zeroes of the Riemann zeta function. Product formula of zeta function, first results on the density of non-trivial zeroes.

Tuesday 18.11 The classical zero-free region for the zeta-function

Wednesday 19.11 The Riemann - von Mangoldt formula for the number of zeroes below given level.

Tuesday 25.11 Riemann - von Mangold formula (continuation). Discussion of the explicite formula an beginning of the proof.

Wednesday 26.11 Completion of the proof of the explicite formula.

Monday 1.12 Some estimates of zeta function in the critical strip. Relation between error estimates for Chebyschev's psi-function and for $\pi(x)-li(x).$

Tuesday 2.12 Relations between non-trivial zeroes of the Riemann zeta function (especially the RH) and the regularity of distribution of primes.

FIN