2

Few days ago I've known (but currently I am not able to understand the answer) a nice problem proposed in MathWoverflow by the user Lviv Scottish Book, that is [1].

Yesterday using Wolfram Alpha online calculator I was doing experiments with the Möbius function $\mu(n)$, see this MathWorld, that is an arithmetic function related with the sine function. I am curious about if the convergence of series involving the idea of the students of the cited center and the Möbius function could be interesting to state some divergent series, or well being convergents. The same idea could be write for the Liouville's function.

Question. Is it possible to deduce convergence of $$\sum_{n=1}^\infty\frac{\mu(n)}{\sqrt{n}}|\sin n|^n\,?\tag{1}$$ Can you to deduce convergence of $$\sum_{n=1}^\infty\frac{\mu(n)}{n^a}|\sin n|^n\tag{2}$$ for a fixed real number $\frac{1}{2}<a<1$? Thanks in advance.

See this code with Wolfram Alpha online calculator

sum mu(n)abs(sin n)^n/sqrt(n), from n=1 to 4000

and thatone

sum mu(n)abs(sin n)^n/n, from n=1 to 500

Since $|\sin(n)|\leq 1$ one has that $\sum_{n=1}^\infty\frac{\mu(n)}{n^{s}}|\sin n|^n$ is convergent for $\Re s>1$, using absolute convergence.

References:

[1] Lviv Scottish Book, Is the series $\sum_n|\sin n|^n/n$ convergent?, MathOverflow (posed on 22.06.2017 by Ph D students of H. Steinhaus Center of Wroclaw Polytechnica).

  • 1
    The MO link is https://mathoverflow.net/questions/282259/is-the-series-sum-n-sin-nn-n-convergent – Gerry Myerson Oct 24 '17 at 09:57
  • 1
    Relevant: https://math.stackexchange.com/questions/823816/is-sum-limits-n-1-infty-frac-sin-nnn-convergent – Gabriel Romon Oct 24 '17 at 10:02
  • Many thanks to Gerry, and to you @GabrielRomon Your reference seems more understandable to my level. –  Oct 24 '17 at 10:08
  • If some user is interested and isn't in the literature that he/she feels free to study it, if after this bounty ends there are no solutions. I say about previous question and similar questions for the Liouville's function, and maybe if it has mathematical meaning also for $n^{it}$ (I add this function because I like the expository article , Don't be seduced by the zeros!, by Andrew Granville); I like also the radical of an integer, see this Wikipedia $\operatorname{rad}(n)$ and $n/\operatorname{rad}(n)$. –  Nov 07 '17 at 16:43
  • @2016 this is a comment to draw your attention about my question, if you want to read my question. I've asked it recently and my bounty ended few minutes ago. Isn't required a response for this comment. Good week. –  Nov 10 '17 at 10:45

0 Answers0