TITLE: Questions on Convergence of Explicit Formulas for $f(x)=\sum\limits_{n=1}^x a(n)$ where $a(n)\in\{\left|\mu(n)\right|,\mu(n),\phi(n),\lambda(n)\}$
QUESTION [2 upvotes]: This question is a follow-on to my earlier question at the following link.
What is the explicit formula for $\Phi(x)=\sum\limits_{n=1}^x\phi(n)$?
This question pertains to the explicit formulas for the following four functions where $\mu(n)$ is the Möbius function, $\phi(n)$ is the Euler totient function, and $\lambda(n)$ is the Liouville function. Also note $M(x)$ is the Mertens function.
(1) $\quad Q(x)=\sum\limits_{n=1}^x\left|\mu(n)\right|\,,\qquad \frac{\zeta(s)}{\zeta(2\,s)}=\sum\limits_{n=1}^\infty\frac{\left|\mu(n)\right|}{n^s}$
(2) $\quad M(x)=\sum\limits_{n=1}^x \mu(n)\,,\qquad \frac{1}{\zeta(s)}=\sum\limits_{n=1}^\infty\frac{\mu(n)}{n^s}$
(3) $\quad\Phi(x)=\sum\limits_{n=1}^x \phi(n)\,,\qquad \frac{\zeta(s-1)}{\zeta(s)}=\sum\limits_{n=1}^\infty\frac{\phi(n)}{n^s}$
(4) $\quad L(x)=\sum\limits_{n=1}^x \lambda(n)\,,\qquad \frac{\zeta(2\,s)}{\zeta(s)}=\sum\limits_{n=1}^\infty\frac{\lambda(n)}{n^s}$

I've read the explicit formulas for the four functions defined above are as follows.
(5) $\quad Q_o(x)=\frac{6\,x}{\pi^2}+\sum\limits_\rho\frac{x^{\frac{\rho}{2}}\,\zeta\left(\frac{\rho}{2}\right)}{\rho\,\zeta'\rho)}+1+\sum\limits_{n=1}^N\frac{x^{-n}\,\zeta(-n)}{(-2\,n)\,\zeta'(-2\,n)}\,,\quad N\to\infty$
(6) $\quad M_o(x)=\sum\limits_\rho\frac{x^{\rho}}{\rho\,\zeta'(\rho)}-2+\sum\limits_{n=1}^N\frac{x^{-2\,n}}{(-2\,n)\,\zeta'(-2\,n)}\,,\quad N\to\infty$
(7) $\quad\Phi_o(x)=\frac{3\,x^2}{\pi^2}+\sum\limits_\rho\frac{x^\rho\,\zeta(\rho-1)}{\rho\,\zeta'(\rho)}+\frac{1}{6}+\sum\limits_{n=1}^N\frac{x^{-2\,n}\,\zeta(-2\,n-1)}{(-2\,n)\,\zeta'(-2\,n)}\,,\quad N\to\infty$
(8) $\quad L_o(x)=\frac{\sqrt{x}}{\zeta\left(\frac{1}{2}\right)}+\sum\limits_\rho\frac{x^\rho\,\zeta(2\,\rho)}{\rho\,\zeta'(\rho)}+1$

The four explicit formulas defined in (5) to (8) above are illustrated in the following four plots in orange and the corresponding reference functions defined in (1) to (4) above are illustrated in blue. All plots are evaluated over the first 200 pairs of zeta zeros and the sum over $n$ is also evaluated with the upper limit $N=200$. The red discrete portions of the plots illustrate the evaluations of the explicit formulas at integer values of $x$.


$\text{Figure (1): Illustration of $Q_o(x)$}$


$\text{Figure (2): Illustration of $M_o(x)$}$


$\text{Figure (3): Illustration of $\Phi_o(x)$}$


$\text{Figure (4): Illustration of $L_o(x)$}$

I initially thought perhaps the formula for $\Phi_o(x)$ was wrong as it seemed to exhibit a different convergence than the the formulas for $Q_o(x)$ and $M_o(x)$ which was the motivation for my earlier question, but I subsequently noticed the formulas for $Q_o(x)$ and $M_o(x)$ also seem to exhibit slightly different convergences. Note $Q_o(x)$, $M_o(x)$, and $\Phi_o(x)$ illustrated in figures (1), (2), and (3) above seem to converge for $x>b$, $x>c$, and $x>1$ respectively where $0<b<c<1$. I'm now trying to understand why explicit formulas such as $Q_o(x)$, $M_o(x)$, and $\Phi_o(x)$ seem to exhibit different lower convergence bounds.
Question (1): Is there a simple explanation as to what determines the lower convergence bound with respect to $x$ of explicit formulas such as $Q_o(x)$, $M_o(x)$, and $\Phi_o(x)$?
Question (2): Is there an explicit formula analogous to those above that actually converges for $x>0$?

Note the explicit formula $L_o(x)$ illustrated in Figure (4) above doesn't seem to converge.
Question (3): Is the explicit formula $L_o(x)$ defined in (8) above incorrect and if so, what is the correct explicit formula for $L(x)$?

REPLY [0 votes]: you would need to introduce a test function $ f(x)$ to make it convergent for example
$$ \sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}g(\log n)=\sum_{\gamma}\frac{h( \gamma)}{\zeta '( \rho )}+\sum_{n=1}^{\infty} \frac{1}{\zeta ' (-2n)} \int_{-\infty}^{\infty}dxg(x)e^{-(2n+1/2)x} $$
Also for the Liouville function  we have
$$ \sum_{n=1}^{\infty} \frac{\lambda(n)}{\sqrt{n}}g(\log n) = \sum_{\gamma}\frac{h( \gamma)\zeta(2 \rho )}{\zeta '( \rho)}+ \frac{1}{\zeta (1/2)}\int_{-\infty}^{\infty}dx g(x) $$
For the Euler-Phi function the explicit formula reads
$$ \sum_{n=1}^{\infty} \frac{\varphi(n)}{\sqrt{n}}g(\log n)= \frac{6}{\pi ^2} \int_{-\infty}^{\infty}dx g(x)e^{3x/2}+ \sum_{\gamma}\frac{h( \gamma)\zeta(\rho/2 )}{\zeta '( \rho)}+\sum_{n=1}^{\infty}\int_{-\infty}^{\infty}\frac{\zeta (-2n-1)}{\zeta ' (-2n)}dx g(x) e^{-x(2n+1/2} $$
for the square-free function
$$\sum_{n=1}^{\infty} \frac{|\mu(n)|}{n^{1/4}}g(\log n)=  \frac{6}{\pi ^2} \int_{-\infty}^{\infty}dx g(x)e^{3x/4}+ \sum_{\gamma}\frac{h( \gamma)\zeta(\rho -1 )}{\zeta '( \rho)}+ \frac{1}{2}\sum_{n=1}^{\infty} \frac{\zeta (-n)}{\zeta ' (-2n)} \int_{-\infty}^{\infty}dx g(x)e^{-x(n+1/4)} $$
here $ g(x) $ and $ H(x)$ form a fourier transform pair , these formulae are generalizatons of possion summation formula
$ \rho = 1/2+i\gamma $
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