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.

### 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 integrity means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

Academic misconduct is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

#### Plagiarism

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

• Although not intended, unintentional plagiarism is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided.
• Intentional plagiarism is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

#### Unauthorised Collaboration

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

#### Impersonation

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

#### Falsiﬁcation

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

#### Use of Unauthorised Materials

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

#### Assisting Others to Commit Academic Misconduct

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.

Further information

While we strive to keep details as accurate and up-to-date as possible, information given here should be regarded as provisional. Individual lecturers will confirm teaching and assessment methods.
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.