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I've deduced this identity involving the Riemann Zeta function $\zeta(s)$ and the Möbius function $\mu(n)$, the zeros and the meaning of how is understood the summation $\sum_{\rho}$ on assumption of the Riemann Hypothesis is explained in [2]. See my details below of this question.

Question 1. Is right that this identity $$\sum_{k=1}^\infty\frac{1}{k(k+1)}\sum_{\rho}\frac{k^\rho}{\rho\zeta'(\rho)}+\sum_{l=1}^\infty\frac{(-1)^{l-1}(2\pi)^{2l}}{(2l)!l\zeta(2l+1)}\sum_{k=1}^\infty \frac{1}{(k+1)k^{2l+1}}=2+\frac{1}{2}\sum_{k=1}^\infty\frac{\mu(k)}{k+1}$$ holds? Thanks in advance.

My deduction was to combine the identity that satisfy the Mertens and Möbius functions, that is (1.8) of page 2 from [1] (currently available in arXiv with article ID arXiv:math/0011254) with the exact formula of the Mertens function, see [2], I am saying the identity (2.3) in page 141, thus we are on assumption of the Riemann Hypothesis and the summation $\sum_{\rho}$ has the meaning (but I employ previous notation) explained by the authors. The simplifications were using the Prime Number Theorem (I know also that RH implies PNT), then from $$-\frac{1}{2}\sum_{k=1}^{n-1}\frac{\mu(k)}{k(k+1)}+\sum_{k=1}^{n-1}\frac{M(k)}{k(k+1)}=\sum_{k=1}^{n-1}\frac{1}{k(k+1)}\left(\sum_{\rho}\frac{k^\rho}{\rho\zeta'(\rho)}\right)-2\sum_{k=1}^{n-1}\frac{1}{k(k+1)}$$ $$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{k=1}^{n-1}\frac{1}{k(k+1)}\sum_{l=1}^\infty \frac{(-1)^{l-1}(2\pi)^{2l}}{(2l)!l\zeta(2l+1)k^{2l}} ,$$

I've calculated using the identity that satisfy the Mertens and Möbius functions with the Prime Number Theorem that $$-\frac{1}{2}\left(0-\sum_{k=1}^\infty\frac{\mu(k)}{k+1}\right)+0-0=\sum_{k=1}^\infty\frac{1}{k(k+1)}\left(\sum_{\rho}\frac{k^\rho}{\rho\zeta'(\rho)}\right)-2$$ $$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{l=1}^\infty\frac{(-1)^{l-1}(2\pi)^{2l}}{(2l)!l\zeta(2l+1)}\sum_{k=1}^\infty \frac{1}{(k+1)k^{2l+1}}.$$

Question 2. A) Is it possible identify the series $$\sum_{k=1}^\infty \frac{1}{(k+1)k^{2l+1}}?$$ B) (Optional, but I would like to see your answer.) Does make sense (does converge) this expression $$\sum_{\rho}\frac{1}{\rho\zeta'(\rho)}\sum_{k=1}^\infty\frac{k^\rho}{k(k+1)},$$ where $\sum_{\rho}$ means the cited limit? Many thanks.

With Wolfram Alpha I can get some examples of A) for simple values of $l$. If you know how derive a closed form or do you know from the literature please tell/refers me. For B) I know the calculation $$ \left| \sum_{k=1}^\infty\frac{k^\rho}{k(k+1)}\right| \leq \sum_{k=1}^\infty\frac{k^{\Re s}}{k(k+1)}=\sum_{k=1}^\infty\frac{1}{k^{1-\Re s}(k+1)}$$ that is convergent if $\Re s<1$, but I don't know if it is possible/known what about the series in Question 2.B).

References:

[1] Báez-Duarte, Arithmetical Aspects of Beurling's Real Variable Reformulation of the Riemann Hypothesis, (2000).

[2] Odlyzko and te Riele, Disproof of the Mertens conjecture, Journal für die reine und angewandte Mathematik (1985)

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  • $\begingroup$ Please to solve A) only is required a yes if your calculations are the same that mine, but is the best if you want/can add yourself remarks as companion of mine if there are some improvement of my calculaitons or something that you want add it. Thanks. $\endgroup$ – user243301 Jan 15 '17 at 22:58
  • $\begingroup$ All users, today I know how to compute inductively a closed-form in terms of particular values of the Riemann Zeta function to solve Question 2.A), thus only is required an answer as proof-verification for Question 1, and also if it's possible Question 2.B). Many thanks. $\endgroup$ – user243301 Jan 23 '17 at 19:12
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For understanding $\sum_\rho$ (sum over the non-trivial zeros) you need to study the analytic properties of $\zeta(s)$ (analytic continuation, functional equation, growth rate) the residue theorem and possibly some complicated estimations for the growth rate of $1/\zeta(s)$ and the density of zeros on the critical strip (on the critical line if assuming RH).

Everything is explained in the books on the Riemann zeta function : Titchmarsh, Apostol, Edwards, Montgomery.. There are also some teachers notes like thisone proving in details the prime number theorem for $\zeta(s)$ and $L(s,\chi)$ (in arithmetic progressions).

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  • $\begingroup$ Many thanks for your attention. Many thanks for your remarks in first paragraph, and references in the second paragraph. My main intention in this post was to know if my calculations were rights and identify the series in Question 2.A). Any case sincerely thanks. $\endgroup$ – user243301 Jan 16 '17 at 6:36

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