Is the following function reducible in any way, even if just for simplifying the computation of it at a single point (e.g. with Taylor series expansion)? I'm also curious if anyone has seen or studied this particular function (or variants of it) before. Does it have a name, for example?

$$ p(x) = -\sin(2\pi x) \sum_{n=2}^\infty \frac{\tan(\pi x/n+\pi/2)}{2n} $$

The function appears to have a value of 1 with a slope of 0 at the primes.

enter image description here

  • $\begingroup$ From a plot, it looks to me as though $p(x)=0$ whenever $x$ is a half-integer (where $\sin 2\pi x$ vanishes). $\endgroup$
    – rogerl
    Commented Dec 12, 2016 at 16:55
  • $\begingroup$ Yes, $\sin(2\pi x)$ is 0 at all half-integers, and more specifically, it is 0 and decreasing at all integers. However, the second part of the product, the infinite sum of tangents, is +/- infinity for all multiples of integers depending on whether the limit is approached from the left or right. But for each $2 <= n <= \infty$, both left and right one-sided limits of the product at every multiple of n are both 1, so the sum over all such products is exactly 1 for primes and more than 1 for non-primes. $\endgroup$
    – fred271828
    Commented Dec 12, 2016 at 18:33


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