ASKSAGE: Sage Q&A Forum - Latest question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 27 Oct 2011 05:01:44 -0500Zeta function gone wild?http://ask.sagemath.org/question/8418/zeta-function-gone-wild/Playing with number fields we reached some weird numerical results. Investigating the problem boiled down to weird output of zeta functions at odd positive integers from 7 onward.
For example,
sage: K.<a> = NumberField(x^2-2)
sage: K.zeta_function()(7)
82.7603619399160
sage: K.zeta_function(prec=100)(7)
45333.379954778857657650185188
sage: K.zeta_function(prec=200)(7)
5.6555192254423051174292272646037247772094677139829119697339e8
These answers seem to be all erroneous. Close values behave fine:
sage: K.zeta_function()(7.0000001)
1.00787667933529
sage: K.zeta_function()(7.00000001)
1.00787667982152
sage: K.zeta_function()(7.000000001)
1.00787669344227
sage: K.zeta_function()(7.0000000001)
1.00787388932573
(the problem begins afterwards).
Another way to obtain this value is
sage: quadratic_L_function__exact(7,2)*zeta(7)
TypeError: n must be a critical value!
which doesn't work. It turns out that this does work:
sage: quadratic_L_function__numerical(7,2)*zeta(7).n()
1.00787667988590
but this does not help us for general number fields (only quadratic).
The same happens with Riemann zeta function, which is the Dedekind zeta function of $\mathbb{Q}$:
sage: K.<a> = NumberField(x)
sage: K.zeta_function()(7)
verbose -1 (371: dokchitser.py, __call__) Warning: Loss of 2 decimal digits due to cancellation
52.5237126027390 # Wrong!
sage: zeta(7).n() # Checking Riemann directly
1.00834927738192 # okie dokie
sage: K.zeta_function()(7.00001)
verbose -1 (371: dokchitser.py, __call__) Warning: Loss of 2 decimal digits due to cancellation
1.00834921704698 # correct
so we get a warning here, and the value at 7 is wrong, but at 7.00001 things are fine.
We are guessing that this relates to zeros/poles of Gamma in the functional equation for zeta. However, since `zeta(7)` and `quadratic_L_function__numerical` do work fine, perhaps there is a way to calculate other zeta functions there as well?
parzanThu, 27 Oct 2011 05:01:44 -0500http://ask.sagemath.org/question/8418/