You are cordially invited to the Number Theory Seminar organized by the Department of Mathematics.
Speaker: Tomos Parry (Bilkent)
“Summation formulas in analytic number theory”
Abstract: Let π(x) be the number of primes up to x. If you want to count primes then you can write the count as an integral in the complex plane
π(x)= ∫_(2-i∞)^(2+i∞)▒〖P(s) x^s/s ds〗
where P(s)=∑_p▒1/p^s .
Analysing the complex analytic properties on the RHS then lets you say something about the arithmetic problem on the LHS – this is the classical proof of the Prime Number Theorem.
The above formula is known as Perron’s formula and is really an example of a Mellin inversion. In this mini seminar series we’ll discuss the formula in this Mellin inversion context and discuss further formulas exemplifying this, especially Voronoi’s summation formula.
The nature of the talks will be informal and the discussion of Perron and Mellin inversion will be more of an overview rather than details, lasting 2-3 talks. This part should give an idea of what analytic number theory looks like from afar – in particular the hope is it can appeal to undergraduates if they know some complex analysis. In the 2-3 talks after we’ll see what it looks like up close by doing Voronoi in more detail.
Date: Friday, Feb 21, 2025
Time: 19:00
Place: SB-Z11