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@MichaelAlbanese the sum of $a_i$ is zero, so adding those $1/n$ is adding zero, but makes the infinite sums converge. The usage of the variable $x$ is traditional in partial fractions decomposition, indicating the equality is true for all, say, reals, and not just integers.
That there is a $\varepsilon>0$ for which we can disprove the statement for all smaller numbers, I think, follows from Cilleruelo's conjecture for the polynomial in question, and specifically we can take $\varepsilon=1/12$.
I think it should possible to find a specific curve over rationals, such that you can show that (up to log factors) a positive proportion of quadratic twists with discriminant supported on the first $n$ primes have root number -1, giving a much higher lower bound than logarithmic