# Limit superior, limit inferior and a series involging $\sum_{k\nmid n}$k, where $1\leq k\leq n$

The purpose of this post is state assertions by the use of statements and hypothesis in an expository way and after I am asking for reasonable unconditionally results that you can provide us.

Using Gronwall's theorem (see for example ), the inequality derived for arithmetic and geometric means, we obtain without an use of Riemann hypothesis

$$\limsup_{n\to \infty}\frac{(n!)^{1/n}}{n\log\log n}\leq e^{\gamma}+\limsup_{n\to \infty}\frac{\delta(n)}{n\log\log n}$$

where $\delta (n)$ is defined as the sum of integers $k$, $1\leq k\leq n$ such that $k\nmid n$; by Gauss statement $\sum_{k=1}^n k=\frac{n(n+1)}{2}$, this arithmetic function $$\delta(n)=\frac{n(n+1)}{2}-\sigma(n)$$ which isn't multiplicative since $\delta(1)=0\neq 1$, and $\gamma$ is the Euler's contant. In the other hand by Gauss statement and Gronwall we compute

$$\infty\leq e^\gamma+\limsup_{n\to \infty}\frac{\delta(n)}{n\log\log n}$$

Thus I believe that there no exists this $\limsup$. Too, I know that for positive quantities $\liminf\leq \limsup$, and $\sigma(n)>n$ implies by Gauss $\delta(n)\geq \frac{(n+1)(n-2)}{2}$. The purpose of following question is refresh notions on $\limsup$ and $\liminf$ and try compute the best unconditionally statements possible, you can use Robin and Erdös refined statements (see )

Question. Can you compute unconditionally or give bounds to $\limsup$ and $\liminf$ in these cases: a) same cited case for $\delta(n)/(n\log\log n)$ and b) same question but you scales previous quotient as $n^{\alpha}(\log\log n)^\beta$, where $\alpha,\beta$ are the constants/functions that you desired/needs. c)(Optional question). If you can/want use Riemann hypothesis to give some statement about $\limsup,\liminf$ they are welcome.

Using Robin's statement for Riemann hypothesis (see ) and Gauss, thus now I use the hypothesis here, it is easy to prove

$$C:=\sum_{n=5041}^\infty\frac{1}{\delta(n)+e^\gamma n\log\log n}<2\sum_{n=5041}^{\infty}\frac{1}{n(n+1)}$$

which is convergent since Leibnitz said that $\sum (\frac{1}{n}-\frac{1}{n+1})$ is telescoping to first term of the sum.

Question. a)Can you give the best unconditionally behaviour of previous series involving $\delta (n)$. b) (Optional question involving a computational experiment). Compute an approximation to constant $C$, with a computer.

I don't know if previous exercises are in literature, yet. Thanks in advance, my goal is learn and y to encourage people, with these easy facts and computations. Of course my power is not solves, but yes learn.

References:

• I'm agree that there are several questions, and it isn't the best, too my attempts were not the best, I will close te prize on the more complete answer this week. – user243301 Oct 28 '15 at 11:53