Mathematics
Te Tari Pāngarau me te Tatauranga
Department of Mathematics & Statistics

## MATH4NT Analytic Number Theory

 Second Semester
10 points

Number theory, which is probably the oldest branch of mathematics, is concerned with properties of integers. Of particular interest are prime numbers. Analytic number theory studies these using methods from analysis. This module gives a basic introduction with a focus on arithmetic functions, asymptotic formulae and the distribution of prime numbers.

### Period

2018, Semester 2.

### Prerequisites

MATH201.

MATH302 is recommended.

### Main topics

#### Fundamentals:

• divisibility, factors, prime numbers
• fundamental theorem of arithmetic
• various proofs of Euclid's theorem

#### Arithmetic functions:

• Dirichlet multiplication
• the Möbius inversion formula
• generalised convolutions
• Legendre's identity

#### Asymptotic formulae:

• big O notation and asymptotic equality
• Euler's summation formula and Abel's identity
• elementary asymptotic formulae
• Shapiro's Tauberian theorem

#### The distribution of prime numbers:

• the prime number theorem
• equivalent formulations
• bounds for the prime-counting function and the nth prime

#### Riemann's zeta function:

• analytic continuation of the zeta function
• functional equation
• trivial and nontrivial zeros

### Lecturer

Dr Jörg Hennig, room 215, email: jhennig@maths.otago.ac.nz

Office hours: by arrangement (or just pop in if I am in my office).

### Assessment

There are three marked assignments and a final exam.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

F = 0.6E + 0.4A

where:

• E is the Exam mark
• A is the Assignments mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

Academic endeavours at the University of Otago are built upon an essential commitment to academic integrity.

The two most common forms of academic misconduct are plagiarism and unauthorised collaboration.

Plagiarism is defined as:

• Copying or paraphrasing another person’s work and presenting it as your own.
• Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
• Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
• Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

The prime-counting function $\pi(x)$ gives the number of primes less than or equal to $x$. While this function is very irregular on small scales (top panel), it looks remarkably smooth on large scales (bottom panel). A central result in analytic number theory is the Prime Number Theorem, which states that $$\lim_{x\to\infty}\frac{\pi(x)}{x/\ln x}=1.$$ Bernhard Riemann found an explicit formula for the prime-counting function $\pi(x)$, which involves a sum over zeros of the Riemann zeta function $\zeta(s)$. The picture below shows a coloured contour plot of the argument of the zeta function in the complex $s$-plane. One can see the pole at $s=1$, some of the so-called “trivial” zeros on the negative real axis, and some of the “nontrivial” zeros with $\Re(s)=1/2$. The zeta function is one of the most important functions in analytic number theory.