Just a short time ago I have asked this question with a trivial answer that I missed: Nature of the prime numbers series: $\sum \limits _{p}\frac{2^p\mod p}{p^2}$ What I am interested in is something like this: Convergence of the series $\sum_ {n\geq1} \frac {(f(n) +P(n)) \pmod {Q(n)}} {D(n)}$ but with n taken for prime numbers. So, my question is: Does $$\sum \limits _{p}\frac{(2^p+p^3)\mod (2*p+5)}{p^2}$$ diverge as an particular example.
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1The expression $\frac{2^p\mod p}{p^2}$ is still somewhat natural , but those expressions look extremely arbitary. I do not think that we can decide the convergence in every case. It might work in the particular case, but I doubt we can establish generalizations able to deal with all such cases. In many cases , we will be able to show that the residue is "often enough" , say , larger than $\frac{p}{3}$ to show the divergence. – Peter Sep 20 '22 at 11:55
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I am thinking that these kind of series are almost always divergent unless some reason the denominator does not provide an constant distribution in 0..n-1, as you have shown in the previous question. Thanks for having fun along me Peter! – Aurelian Florea Sep 20 '22 at 12:09
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@Peter and Peter, by the way, there is also this: https://math.stackexchange.com/questions/4535181/follow-up-question-on-the-nature-of-the-sum-limits-n-geq-1-frac2n-mod-n which when taken only for primes instead off all naturals' reciprocals is when all hell breaks loose. I think then you can only decide on case-by-case basis or who knows if at all. – Aurelian Florea Sep 20 '22 at 15:00