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 [1]), 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 [1])

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 [1]) 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.


[1] http://mathworld.wolfram.com/DivisorFunction.html

  • $\begingroup$ 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. $\endgroup$ – user243301 Oct 28 '15 at 11:53

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