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Solving zeta function equation numerically

asked 2016-05-31 18:07:50 -0600

ec92 gravatar image

I'm trying to solve the equation $$ \zeta'(x)/\zeta(x) = - 3/4 $$ numerically. I'm expecting/hoping for an answer between 1 and 10.

I tried

$\texttt{ find_root(diff(zeta(x))/zeta(x) + 0.75, 2, 40)}$,

but this returns 0.0. I don't know what that ratio of zeta functions looks like, so it's possible there isn't a root, but I don't see why I'm getting an answer outside the specified interval.

Can anyone help find the true root? Thanks.

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What version of Sage are you using? I get the following in Sage 7.2:

sage: find_root(diff(zeta(x))/zeta(x)+.75, 2, 40)
Traceback (most recent call last):
...
RuntimeError: f appears to have no zero on the interval
slelievre gravatar imageslelievre ( 2016-05-31 23:12:21 -0600 )edit

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answered 2016-05-31 20:04:24 -0600

If you plot your function first, for example with

f = diff(zeta(x))/zeta(x)+.75
plot(f,0,10,ymin=-5,ymax=5,detect_poles=True,figsize=[4,3])

you can see approximately where the zero is:

image description

You just need to change the range over which you're searching:

find_root(f,0,10)
1.8333259941894324

Here's a link to live example.

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Asked: 2016-05-31 18:07:50 -0600

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Last updated: May 31 '16