You are cordially invited to the Number Theory Seminar organized by the Department of Mathematics.
“Voronoi’s summation formula”
Speaker: Tomos Parry (Bilkent)
Abstract: The divisor function
d(n):=# divisors of n
is one of the most studied sequences in number theory. This is the fourth seminar about d(n) in a mini-series, where so far we have:
- Discussed Perron’s formula – a contour integral expression for whatever we’re counting. This is the starting point of studying a sequence through the analytic properties of a series associated to the sequence (its Dirichlet series). This let us approximate the average of d(n) up to an error √x (which is no better than an elementary argument though).
- Analysed the integrals in Perron’s formula more carefully – improving the error to x^(1/3), which is now considerably better than elementary.
- Kept the integrals explicit and wrote the error as a sum of these terms – this is Voronoi’s summation formula.
This week is 4. Repeat the above analysis but with a smoothing weight – this shows us that the integral analysis wasn’t really the “important part” of the above arguments, rather just Mellin inversion and continuation of the Dirichlet series.
Date: Friday, March 21, 2025
Time: 19:30
Place: SB-Z11