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I would like to ask this question to know if it is possible to say something about the asymptotic of next sequence, that seems erratic. I was inspired in a sequence showed in this post of Mathematics Stack Exchange, my sequence thus is this variation

$$\sum_{k=1}^n\sin\left(N_k\right),\tag{1}$$ where $N_k=\prod_{j=1}^kp_j$ is the primorial of order $k$ (see the Wikipedia's page dedicated to Primorial), and thus $p_j$ denotes the $j$th prime number.

Question. Prove or refute that: Exist a positive constant $C$ such that for each sufficiently large integer $n$, $$\left|\sum_{k=1}^n\sin\left(N_k\right)\right|\leq C\sqrt{n}\log\log n\tag{2}$$ holds. Many thanks for your attention.

I think that using the Vinogradov notation previous claim is that: $$\sum_{k=1}^n\sin\left(N_k\right)\ll\sqrt{n}\log\log n,$$ holds as $n\to\infty$.

If this question was in the literature please answer this as a reference request and I try to search and read such theorem from the literature.

I think that maybe is interesting to know what work can be done and by means what methods of reasonings, but as was said the sequence seems erratic. Can you refute my claim?

Computational fact. I've created this conjecture using few experiments, you can use next code to make comparisons. The code is the line

for (n = 1, 500, print(sum( k=1, n, sin(prod( j=1, k, prime(j))))))

choose GP as language from this Sage Cell Server and press Evaluate.

  • I like also the answer of Terence Tao for the MathOverflow post Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$? Maybe is interesting try the same reasoning of Terence Tao for different arithmetic functions. I don't know if the case involving the arithmetic function $a(n)=p_n$, the $n$th prime number is feasible. –  Feb 28 '18 at 22:18
  • To me is more nice the bound $n^{\frac{1}{2}+\epsilon}$ in $(2)$ $\forall\epsilon>0$, but from my few experiments and knowledges I have no evidence for this. –  Mar 08 '18 at 09:33

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