In the past few years I have become interested on a type of questions I have stumbled upon. Here are some posts that have produced various results: Problem 1 Basic example with exponential, Problem 2 Basic example with polinoial,Problem 3 General example,Problem 4 Prime Numbers series,Problem 5 with denominator multpiplied by a loghartihm. All these are series where the denominator is a polinomial (times a logharithm on problem 5) and the numerator contains a certain function to which a modulus operation is applied to respect to another polinomial of a degree smaller by one in comparison to the denominator. Let us focus a bit on this series: $$\sum_ {n = 1}^\infty \frac {(f(n) +P(n)) \pmod{ Q(n)}} {D(n)}$$, which is the third problem discussed above. This series heuristically diverges. I have tried now more restrictive conditions to try to break the heuristic. I'd like to discuss the following series: $$\sum_ {n = 1}^\infty \frac {(f_1(n) +P_1(n)) \pmod{ Q_1(n)}(f_2(n) +P_2(n)) \pmod{ Q_2(n)}} {D(n)}$$, where $$Q_1(n)≠Q_2(n)$$ and $$deg(Q_1(n))+deg(Q_2(n))+1=deg(D(n))$$ I have worked out, that under the assumption that $Q_1(n)$ and $Q_2(n)$ are equidistributed, and assuming $Q_1(n)<Q_2(n)$, starting with a large enough n, there is a weaker heuristic that still holds, and that the series is still divergent. We assume that on average the remainder from $Q_1(n)$ yields a expected value of $(Q_1(n)-1)/2$, and also the remainder from $Q_2(n)$ yields a expected value of $(Q_2(n)-1)/2$. From there we obtain, after some calculations that the expectation of the full numerator is $\frac {(Q_1(n)-1)*(Q_2(n)-2)}{4}$, which still gives an divergent series. Am I correct here? I am more interested in a discussion as otherwise the problem is very hard to solve.
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