# On the series $\sum_{n=1}^{\infty} (H_{n}+\exp(H_{n})\log(H_{n}))/n^{s}$, where $H_{n}$ is the $n$th harmonic number

It is known the following (see [1], here is an open access PDF on his homepage):

Theorem (Lagarias, 2002). Let $\sigma(n)$ denote the sum of the positive divisors of $n$. Riemann Hypothesis holds if and only if $$\sigma(n)\leq H_n + \exp( H_n)\cdot \log( H_n)$$ for every $n$, where $H_n= 1+1/2+\cdots +1/n$.

On the other hand, let $\zeta$ be the Riemann function, it is known that for $\sigma >2$, see [2] page 229, (and page 231 for its corresponding Euler product if you need it) $$\zeta (s)\zeta(s-1)=\sum_{n=1}^{\infty}\frac{\sigma(n)}{n^{s}}$$ converges absolutely, where here $s=\sigma + it$ is the complex variable of this Dirichlet series, and thus $\sigma$ its real part.

Thus, we try to obtain a relationship between previous information as, first multiply Lagarias' inequality by $n^{-s}$, and secondly we obtain a series when add all terms, and finally we assume $\sigma >2$, this is $$\zeta (s)\zeta(s-1)\leq \sum_{n=1}^{\infty} \frac{(H_{n}+\exp(H_{n})\log(H_{n}))}{n^{s}}$$

I ask to Math Stack Exchange community this

Question. What about absolute convergence of the series on right term in previous inequality? And, can we obtain some information about the statement of the unsolved problem, the so called Riemann Hypothesis, from previous inequality (it is previous inequality has a significative sense, or not, is not useful)?

Thanks, I don't know if there is previous literature about this specific question, neither my question is really useful; I edit this question because if from the discussion about convergence of this series I can learn at same time, I accept that second part in my question is speculative, but I want show to you if you want think about it.

References:

[1] 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.

[2] Tom M. Apostol, Introduction to Analytic Number Theory, Springer (1976).

• Thanks for your edit Zev Chonoles. – user243301 Aug 2 '15 at 23:48

I'm assuming that that the series on the RHS is $$\sum_{n\geq1}\frac{H_{n}+\exp\left(H_{n}\right)\log\left(H_{n}\right)}{n^{s}}.$$ It's well known that $$H_{n}=\log\left(n\right)+\gamma+o\left(1\right)$$ where $\gamma$ is the Euler-Mascheroni constant. Hence the series is absolutely convergent if $\textrm{Re}\left(s\right)>2$. About the second question, there is no reason to belive that this is in some way useful for the Riemann hypothesis. Remember that the identity $$\zeta\left(s\right)=\sum_{n\geq1}\frac{1}{n^{s}}$$ holds only for $\textrm{Re}\left(s\right)>1$.
• It's a nice answer, I replaced your asymptotic and prove that really this provide the absolute conergence for $\sigma >2$, so this is what I learn . I accept your argument para the second part, I had not counted with the condition of convergence that say . Very thanks much you @MarcoCantarini – user243301 Aug 3 '15 at 6:05