[seqfan] Re: simplify the constant A186706

israel at math.ubc.ca israel at math.ubc.ca
Wed Jul 25 20:28:07 CEST 2012

On Jul 25 2012, Alexander P-sky wrote:

>WolframAlpha reports that
>Integrate[DedekindEta[x I], {x, 0, Infinity}] -
>~= 1.53825*10^(-10)

That's only to 10 digits of accuracy.

Use the definition of DedekindEta as a sum:

Eta(i x) = sum_{n=-infinity}^infinity (-1)^n exp(-pi x (6n-1)^2/12)

Now int_0^infinity exp(-pi x (6n-1)^2/12) dx = 12/(pi (6n-1)^2)

According to Maple, sum_{n=-infinity}^infinity (-1)^n 12/(pi (6n-1)^2) is


It won't simplify the difference between this and 2 pi/sqrt(3) to 0, but 
floating point evaluation at 1000 digits gives -.1e-998+0.*I. So it seems 
very likely that these are equal.

Robert Israel
University of British Columbia

More information about the SeqFan mailing list