MATH4NT Analytic Number Theory
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.
2018, Semester 2.
MATH302 is recommended.
- divisibility, factors, prime numbers
- fundamental theorem of arithmetic
- various proofs of Euclid's theorem
- Dirichlet multiplication
- the Möbius inversion formula
- generalised convolutions
- Legendre's identity
- 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
Office hours: by arrangement (or just pop in if I am in my office).
There are three marked assignments and a final exam.
Your final mark F in the paper will be calculated according to this formula:
F = 0.6E + 0.4A
- 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.
Academic misconduct: Plagiarism
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.
Academic misconduct: 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 student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.