# A mixture with ingredients of two equivalences with Riemann Hypothesis

Let $f(x)=x\cdot(\log x)^x$ for $x\geq 2$, then integrating $\log f(x)=\int_2^x f'(t)/f(t)dt$, it is easy to prove the first statement of following, and directly if we put $x=e^H_n$ and add $H_n$ the second ($e>2$)

Lemma. For $x\geq 2$, $$x\log \log x=2\log\log 2 +Li(x)+\int_2^x\log\log tdt$$ where $Li(x)$ is the logarithmic integral $\int_2^xdt/\log t$. And thus $$H_n+e^{H_n}\log (H_n)=H_n+2\log\log 2+Li(e^{H_n})+\int_2^{e^{H_n}}\log\log tdt$$ where $H_n$ is the nth harmonic number $1+1/2+\cdots +1/n$.

Now my question is

Question. Can you continue previous computations to prove a reasonable improve of $$\sigma(n)\leq \text{something}\leq \text{RHS of second statement in previous lemma},$$ where $\sigma(n)$ denote the sum of the positive divisors of $n$, $\sum_{d|n}d$.

Thanks is advance, my only goal is edit the best possible post in this Math Stack Exchange, I apologize to bring this question here because I believe that this community can made contributions. Sorry by my english.

Appendix:

My question as is built, tell us how take an ingredient from an equivalence of Riemann Hypothesis, $\sigma(n)$, and other ingredient, $Li(x)$, from another equivalence of RH and put these together, well, in a not understandable way. Topics as certain type of abundant numbers, Robin approach to RH, the order of growth of $\sigma(n)$ or $H_n$, and the relationship between $Li(x)$ and Riemann Hypothesis can find in the literature. In  (I have not an open source to provide it) we can read an integral equal to $\sigma(n)$.

Simply you can read the first paragraph of the section Initial Ideas, page 343 of  (given as free paper), and Lagarias equivalence in the last paragraph before the section Other Zeta and L-functions in page 347 to know two equivalences with Riemann Hypothesis.

Thus (via ) Riemann Hypothesis holds if and only if $$\sigma(n)\leq H_n+2\log\log 2+Li(e^{H_n})+\int_2^{e^{H_n}}\log\log t dt.$$

References:

 Wikipedia.

 J. Brian Conrey, The Riemann Hypothesis, Notices of the AMS, MARCH 2003, Volume 50, Number 3 (www.ams.org/notices).

 Jeffrey C. Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis, The American Mathematical Monthly Vol. 109, No. 6 (Jun. -Jul., 2002), pp. 534-543.

 George Purdy, An Integral Equal to $\sigma(n)$, Problems and Solutions, Problem E 1850 [1966, 82] American Mathematical Monthly Vol. 74 N. 5 MAY 1967, p. 594-595.

• I've made the change in the misprint and deleted the first comment, to avoid confusion, it is better . Thanks. – user243301 Sep 10 '15 at 6:45

$\int \limits_{2}^{x} \ln \ln t dt = t \ln \ln t-Li(t)|_{2}^{x} = x \ln \ln x-2 \ln \ln 2+Li(2)-Li(x)$
So $\int \limits_{2}^{x} \ln \ln t dt+ 2 \ln \ln 2+Li(x)+\ln x = \ln x+ Li(2)+x \ln \ln x$, put $x=e^{H_n}$ to get $H_n + Li(2) +e^{H_n} \ln H_n$
So we arrive at $Li(2) + H_n +e^{H_n} \ln H_n \geq \sigma(n)$ without the $Li(2) \approx 1.04516$ its the famous C.Lagarias inequality which is true for all $n\geq 1$ iff the R.H. is true.