1
$\begingroup$

Let $\mu(n)$ the Möbius function, see the definition from this MathWorld. We consider the series $$\sum_{n=1}^\infty\sum_{k=1}^\infty\frac{\mu(n)\mu(k)}{n^k k^n}.\tag{1}$$ I would like to know how to prove that this series is convergent.

My attempt was to use absolute convergence, but it fails: $$ \left|\sum_{n=1}^\infty\sum_{k=1}^\infty\frac{\mu(n)\mu(k)}{n^k k^n} \right|\leq \sum_{n=1}^\infty\sum_{k=1}^\infty\frac{1}{n^k k^n},$$ and $\sum_{n=1}^\infty\sum_{k=1}^\infty\frac{1}{n^k k^n}$ is divergent, since (this last series has positive terms) $$\sum_{n=1}^\infty\sum_{k=1}^\infty\frac{1}{n^k k^n}=\left(\sum_{n=1}+\sum_{n=2}^\infty\right)\sum_{k=1}^\infty\frac{1}{n^k k^n}=\sum_{k=1}^\infty\frac{1}{ k}+\sum_{n=2}^\infty\sum_{k=1}^\infty\frac{1}{n^k k^n}.$$

As motivation, but it is mainly curiosity, I've created $(1)$ when while reading [1] and I tried to set variations of those identities, calculating approximations with Wolfram Alpha, of series like than previous, or $$\sum_{k=1}^\infty\sum_{j=1}^k\frac{\mu(j)}{kj(k+j)}$$ or $$\sum_{k=1}^\infty\sum_{k=1}^k\frac{\mu(k)\mu(j)}{kj(k+j)}.$$ I don't know if such series were in the literature, or are interestings.

Question. Can you prove that $$\sum_{n=1}^\infty\sum_{k=1}^\infty\frac{\mu(n)\mu(k)}{n^k k^n}$$ does converge? Many thanks.

One can see my approximation with Wolfram Alpha online calculator if type the code

sum mu(n)mu(k)/(k^n n^k), n=1 to 30, from k=1 to 30

or

sum mu(n)mu(k)/(k^n n^k), n=1 to 100, from k=1 to 100.

References:

[1] Whalter Janous, Around Apéry's Constant, Journal of Inequalities in Pure and Applied Mathematics, volume 7, issue 1, article 35 (2006).

$\endgroup$
  • 2
    $\begingroup$ Hint: analyze $\sum_{k\geq1}\frac{\mu(k)}{k}$. $\endgroup$ – Marco Cantarini Jul 15 '17 at 10:06
  • $\begingroup$ Many thanks @MarcoCantarini I am going to think in your hint. I know that were in the literature upper bounds for $ \left| \sum_{k\leq x}\frac{\mu(k)}{k} \right|$. Any case feel free to add hints or your approach as an answer. $\endgroup$ – user243301 Jul 15 '17 at 10:17
  • $\begingroup$ I believe @MarcoCantarini is hinting at something like the Riemann zeta function... $\endgroup$ – Simply Beautiful Art Jul 15 '17 at 11:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy