Does the series $\sum \frac{\sin p}{p}$, where $p$ prime, converge?

Asked Jun 17 '18 at 14:38

Active Jun 18 '18 at 07:34

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Question. Assume that $\{p_n\}$ is the sequence of prime numbers, in increasing order. Does the series $$ \sum_{n=1}^\infty \frac{\sin p_n}{p_n} $$ converge?

The only criterion to establish convergence that I can think of, which might be of any use, is the Abel's Test, which requires that the partial sums of the sequence $b_n=\sin p_n$ are bounded, i.e., that there exists an $M>0$, such that $$ |\sin p_1+\cdots+\sin p_n|\le M, \quad\text{for all $n\in\mathbb N$}. $$

Any ideas?

edited Jun 18 '18 at 07:34

asked Jun 17 '18 at 14:38

Yiorgos S. Smyrlis

83,933

1

I think you could use Dirichlet's test here: https://en.wikipedia.org/wiki/Dirichlet%27s_test – Sheel Stueber Jun 17 '18 at 14:41
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See https://math.stackexchange.com/questions/2272248/regarding-the-sum-sum-p-textprime-sin-p?rq=1 – Robert Z Jun 17 '18 at 14:48

Does the series $\sum \frac{\sin p}{p}$, where $p$ prime, converge?

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