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Zeta function gone wild?

asked 2011-10-27 05:01:44 -0500

parzan gravatar image

updated 2011-10-31 05:59:10 -0500

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)
sage: K.zeta_function(prec=100)(7)
sage: K.zeta_function(prec=200)(7)

These answers seem to be all erroneous. Close values behave fine:

sage: K.zeta_function()(7.0000001)
sage: K.zeta_function()(7.00000001)
sage: K.zeta_function()(7.000000001)
sage: K.zeta_function()(7.0000000001)

(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()

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:, __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:, __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?

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This looks like a bug -- have you filed a ticket?

niles gravatar imageniles ( 2011-10-31 09:59:47 -0500 )edit

1 answer

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answered 2014-06-29 11:08:49 -0500

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Asked: 2011-10-27 05:01:44 -0500

Seen: 418 times

Last updated: Jun 29 '14