# How to compute Dedekind eta function efficiently?

According to wiki: https://en.wikipedia.org/wiki/Dedekind_eta_function, Dedekind eta function is defined in many equivalent forms. But none of them is an explicit description (say in algorithmic format) on how to computing it. Where to find such one? Thanks!

• There are some efficient Mathematica programs described in a post on Mathematica.SE May 5, 2018 at 4:28
• Thanks! But it seems there is no an explicit description on how to compute Dedekind eta function on that page. May 5, 2018 at 6:18
• is this helpful for you? (also includes Pari code): Evaluation of the Dedekind eta function May 5, 2018 at 7:59
• Yes, it is very useful! Thanks a lot! May 5, 2018 at 9:06
• @LichengWang Hi. Did you download the PARI/GP tarball which has the complete source code? May 5, 2018 at 18:40

Euler's formula

$$\sum\limits_{n \in \mathbb{Z}} {( - 1)^n q^{\frac{{(3n^2 - n)}} {2}} } = \prod\limits_{n = 1}^\infty {(1 - q^n ),}$$

(which can be proven from Jacobi’s triple product identity by using the fact that $\prod\limits_{n = 1}^\infty {(1 - q^{3n} )(1 - q^{3n - 2} )} (1 - q^{3n - 1} ) = \prod\limits_{n = 1}^\infty {(1 - q^n )}$) provides a good way of numerically computing

$$\eta (\tau )=e^{\frac {\pi {\rm {{i}\tau }}}{12}}\prod _{n=1}^{\infty }(1-e^{2n\pi {\rm {{i}\tau }}})=q^{\frac {1}{24}}\prod _{n=1}^{\infty }(1-q^{n}).$$

My maple code of Gatteschi-Sokal algorithm for computing $R(t,x)=\prod_{n=1}^{\infty}(1-tx^n)$:

GS:=proc(t,x,prec)

local R0, a0, b0, Rn, an, bn, d, c, i, N, r, Rd;

N := 100;

d := 1/2; if d = t*x then d := (1/2)*d end if;

r := evalf$[prec]$(1+d/(1-x));

a0 := 1; b0 := evalf$[prec]$(d/(d-t*x));

R0 := evalf$[prec]$(r*a0+(1-r)*b0);

i := 0;

while i < N do

c := evalf$[prec]$(a0*(d*a0+(1.0-d)*b0));

an := evalf$[prec]$(c/b0);

bn := evalf$[prec]$(c/(x*a0+(1.0-x)*b0));

Rn := evalf$[prec]$(r*an+(1.0-r)*bn);

Rd := evalf$[prec]$(abs(Rn-R0));

if Rd < 10^(-prec) then i := N else

a0 := an; b0 := bn; R0 := Rn; i := i+1 end if;

end do;

return Rn;

end proc;

eta:=(t,prec)->evalf$[prec]$(GS(1,exp($2*Pi*I*t$),prec));

Dedekind_eta := (t, prec)-> evalf$[prec]$(exp($1/12*Pi*I*t$)*eta(t, prec));

Test:

t:=0.3*I:

eta(t,40); Dedekind_eta(t,40);

0.8251926470787677741036466781518992636742

0.7628619270903183863013294748250092216042

Almost same with PARI/GP output:

eta(0.3*I,0)=0.82519264707876777410364667815189926367

eta(0.3*I,1)=0.76286192709031838630132947482500922160