Special and Limiting Values of the Dedekind Eta
- Slides: 29
Special and Limiting Values of the Dedekind Eta Function Neil Bickford and R. W. Gosper or, for Fun and Profit AMS Special Session on Special Functions and Q-Series January 9 th, 2016
The Joy of η for Held group for one partcan of the solution So If we can compute η, the we compute special values of K! surprisingly difficult!
Notation •
The Chowla-Selberg formula (Kaneko(1949, (1990), 1967) Nakkajima and Taguchi’s 1991 version) For instance, the two reduced quadratic forms of discriminant -15 are [1, 1, 4] and [2, 1, 2], so Chowla-Selberg gives (after a bit of simplification) But this doesn’t give individual values (Unless d is a Heegner number. ) Solution: Find quotients of special values!
Weber by example •
Weber by example
Limiting values of η •
Double, double, boil and eta •
Weber by Jacobi …and sometimes the points overlap.
Weber by Jacobi
The bad news •
Spooky action in class field theory d doublings h(d)
The bad news •
Generalizing modular equations (William B. Hart, 2008) •
Generalizing modular equations (William B. Hart, 2008) •
Generalizing modular equations (William B. Hart, 2008) •
Generalizing modular equations (William B. Hart, 2008) • -4 -3 -2 -1 0 1 2 3 4
Generalizing modular equations (William B. Hart, 2008) • -4 -3 -2 -1 0 1 2 3 4
Generalizing modular equations (William B. Hart, 2008) • -4 -3 -2 -1 0 1 2 3 4 which works!
Exploiting modular equations •
Exploiting modular equations • 1 2 3 4
Exploiting modular equations • 1 2 3 4
We now have all the pieces we need to compute the value of η.
Dedekind η for fun and profit d=-51 • • • Use Chowla-Selberg to get a product of η values. Search through quadratic surds to find a way of linking the arguments of η: Find a modular equation that allows you to compute the value of η(z)/η(mz); • Plug in values and solve the resulting polynomial equation… • And substitute back into the original equation to get…
Dedekind η for fun and profit And it works!
evaluate an eta today! Source code available online at http: //bit. ly/eta 2016
Thank you! Neil Bickford nbickford. wordpress. com nbickford@g. ucla. edu Bill Gosper gosper. org billgosper@gmail. com
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