SISSA/IGAP/SUSTech Lectures on "Standard and less standard asymptotic methods" Lecture 4
Channel: ICTP Mathematics
Date: 03/03/2022

Lecturer: Don Zagier

In every branch of mathematics, one is sometimes confronted with the problem of evaluating an infinite sum numerically and trying to guess its exact value, or of recognizing the precise asymptotic law of  formation of a sequence of numbers {A_n} of which one knows, for  instance, the first couple of hundred values.  The course will tell a number of ways to study both problems, some relatively standard (like the Euler-Maclaurin formula and its variants) and some much  less so, with lots of examples.   Here are three typical examples: 1. The slowly convergent sum  sum_{j=0}^\infty (\binom{j+4/3}{j})^{-4/3}   arose in the work of a colleague.  Evaluate it to 250 decimal digits. 2.  Expand the infinite sum  \sum_{n=0}^\infty (1-q)(1-q^2)...(1-q^n) as  \sum A_n (1-q)^n, with first coefficients 1, 1, 2, 5, 15, 53, ... Show numerically that  A_n  is asymptotic to  n! * a * n^b * c  for some real constants a, b and c, evaluate all three to high precision, and recognize their exact values. 3.  The infinite series  H(x) = \sum_{k=1}^\infty sin(x/k)/k  converges for every complex number x.  Compute this series to high accuracy when x is a large real number, so that the series is highly oscillatory.


2022 02 24 MTH Zagier Series L04