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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:

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.