Tools, Techniques, and Algorithms
- Euler-Maclaurin summation.
- The explicit formula. Applications: finding/verifying zeros, density of zeros, correlation of zeros.
- Smooth approximate functional equations. Controlling cancellation.
The Riemann Siegel formula. Schonhage T^{3/8} algorithm for the main sum. Ghaith improvements.
- Incomplete Gamma function and incomplete integrals. Work needed here- improved implementation/algorithms for multiple Gamma factors, and a better understanding of the convergence of the continued fractions used, even for the incomplete Gamma function.
- Bounds for L-functions.
- Convexity.
Odlyzko-Schonhage algorithm. It would be nice to have a publicly available implementation of this, and for more than just \zeta.
- FFT techniques.
- Looking for and counting zeros.
- Booker thesis.
Software
- lcalc (Rubinstein)
- computeL (Dokchitser)
- sympow (Watkins)
pari \zeta_K
- Belabas project (find out more about this)
ZetaPack project (Broughan)
Algorithm to compute from the first zeroes of an L-function primes in a congruence class ?
Data
- Rubinstein: zeros of degree 1 and 2 L-functions, central values of quadratic twists
- Odlyzko: zeros of zeta at large height
- Watkins: symmetric power central values, quadratic twists of E
- Rumely data
- Stein-Watkins BSD data
- Cremona central critical values
- Class number tables
Broughan:Phase portraits of zeta/xi zeros
We had a bit of a discussion concerning indicating the level of rigour used in obtaining the various datasets.