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This question is related to my previous question at the following link.

Does this explicit formula for the prime-counting function $\pi(x)$ converge?


My previous question utilized an explicit formula for Mertens function to derive an explicit formula for the prime-counting function $\pi(x)$. I haven't been able to find a definitive statement as to whether the explicit formula for Mertens function has been proven, and even if so it doesn't seem to be valid all the way down to $x=0$. This limits the number of terms that can be evaluated in the derived explicit formula for $\pi(x)$ requiring the derived formula to be evaluated piece-wise where the evaluation limit changes at integer values of $x$ resulting in discontinuities at integer values of $x$.


This motivated me to investigate an approach based on well-known proven explicit formulas which allow the evaluation limit for the derived explicit formula for $\pi(x)$ to be increased towards infinity. This allows a single evaluation limit to be used over an entire range of $x$ values under evaluation which avoids discontinuities associated with piece-wise evaluation resulting in a continuous function across the entire range of $x$ values under evaluation.


The approach outlined in this question can be applied to many functions of the form $f(x)=\sum_{n=1}^x a(n)$, but this question uses the prime-counting function $\pi(x)$ as an example of $f(x)$. Two explicit formulas for $\pi(x)$ are defined and illustrated. The first explicit formula for $\pi(x)$ is based on von Mangoldt's explicit formula for the second Chebyshev function $\psi(x)$, and the second explicit formula for $\pi(x)$ is based on Riemann's explicit formula for the prime-power counting function $\Pi(x)$.


This question assumes the following definitions. The offset of $6$ below was chosen to avoid a transition of $g_o(x)$ and $h_o(x)$ at $x=0$ which I believe would have adversely affected the convergence of the derived formulas for $\pi_o(x)$ in (10) and (18) below. The offset of $6$ also guarantees a non-zero value of $r(1)$ and $u(1)$ which is necessary in order to derive the Dirichlet inverses below.


(1) $\quad\pi(x)=\sum\limits_{n=1}^x a(n)\,,\quad a(n)=\begin{array}{cc} \{ & \begin{array}{cc} 1 & n\in \mathbb{P} \\ 0 & \text{True} \\ \end{array} \\ \end{array}$

(2) $\quad b(n)=\sum\limits_{d|n} a(d)\,\mu\left(\frac{n}{d}\right)\qquad\qquad\qquad\qquad\qquad\text{(Moebius Transform)}$


(3) $\quad \psi(x)=\sum\limits_{n=1}^x\Lambda(n)$

(4) $\quad g(x)=\psi(x+6)-\log(60)=\sum\limits_{n=1}^x\Lambda(n+6)$

(5) $\quad\psi_o(x)=x-\sum\limits_\rho\frac{x^\rho}{\rho}-\log(2\,\pi)+\sum\limits_n\frac{x^{-2\,n}}{2\,n}\\$ $\qquad\qquad\quad\,\,=x-\sum\limits_\rho\frac{x^\rho}{\rho}-\log(2\,\pi)-\frac{1}{2}\,\log\left(1-\frac{1}{x^2}\right)$

(6) $\quad g_o(x)=\psi_o(x+6)-\log(60)$


(7) $\quad r(n)=\sum\limits_{d|n}\Lambda(d+6)\,\mu\left(\frac{n}{d}\right)\qquad\qquad\qquad\qquad\text{(Moebius Transform)}$

(8) $\quad s(n)=\begin{array}{cc} \{ & \begin{array}{cc} \frac{1}{r(1)} & n=1 \\ \frac{1}{r(1)}\sum\limits_{d|n\land d<n} r\left(\frac{n}{d}\right)\,s(d) & \text{True} \\ \end{array} \\ \end{array}\quad\text{(Dirichlet Inverse)}$

(9) $\quad t(n)=\sum\limits_{d|n} s(d)\,b\left(\frac{n}{d}\right)\qquad\qquad\qquad\qquad\qquad\text{(Dirichlet Convolution)}$


(10) $\quad\pi_o(x)=\sum\limits_{j=1}^J t(j)\,g_o(\frac{x}{j})$


(11) $\quad \Pi(x)=\sum\limits_{n=2}^x\frac{\Lambda(n)}{\log(n)}$

(12) $\quad h(x)=\Pi(x+6)-\frac{7}{2}=\sum\limits_{n=1}^x\frac{\Lambda(n+6)}{\log(n+6)}$

(13) $\quad\Pi_o(x)=li(x)-\sum\limits_\rho Ei(\log\,(x)\,\rho)-\log(2)-\sum\limits_n Ei(-2\,n\,\log(x))\\$ $\qquad\qquad\qquad=li(x)-\sum\limits_\rho Ei(\log(x)\,\rho)-\log (2)+\int_x^\infty\frac{1}{t\,\left(t^2-1\right)\log(t)}\,dt$

(14) $\quad h_o(x)=\Pi_o(x+6)-\frac{7}{2}$


(15) $\quad u(n)=\sum\limits_{d|n}\frac{\Lambda(d+6)}{\log(d+6)}\,\mu\left(\frac{n}{d}\right)\qquad\qquad\qquad\qquad\quad\text{(Moebius Transform)}$

(16) $\quad v(n)=\begin{array}{cc} \{ & \begin{array}{cc} \frac{1}{u(1)} & n=1 \\ \frac{1}{u(1)}\sum\limits_{d|n\land d<n} u\left(\frac{n}{d}\right)\,v(d) & \text{True} \\ \end{array} \\ \end{array}\quad\text{(Dirichlet Inverse)}$

(17) $\quad w(n)=\sum\limits_{d|n} v(d)\,b\left(\frac{n}{d}\right)\qquad\qquad\qquad\qquad\qquad\text{(Dirichlet Convolution)}$


(18) $\quad\pi_o(x)=\sum\limits_{j=1}^J w(j)\,h_o(\frac{x}{j})$


The following two figures illustrate the formulas for $\pi_o(x)$ defined in (10) and (18) above (orange curves) where the formulas for $\pi_o(x)$ are evaluated with an upper limit of $J=25$ and the underlying explicit formulas for $\psi_o(x)$ and $\Pi_o(x)$ defined in (5) and (13) above are both evaluated over the first $1,000$ pairs of non-trivial zeta-zero pairs. The underlying blue reference function corresponds to $\pi(x)$ defined in (1) above. The red discrete portions of the plots in the figures below represent the evaluation of the formulas for $\pi_o(x)$ at integer values of $x$.


Illustration of Formula (10)

Figure(1): Illustration of Formula (10) for $\pi_o(x)$ (orange curve)


Illustration of Formula (18)

Figure(2): Illustration of Formula (18) for $\pi_o(x)$ (orange curve)


Question (1): Do the formulas for $\pi_o(x)$ defined in (10) and (18) above truly converge as $J\to\infty$ and as the number of non-trivial zeta-zero pairs evaluated in the underlying explicit formulas is increased towards $\infty$?


The following three figures illustrate formulas (10) and (18) above for $\pi_o(x)$ eliminate the discontinuities of the previous formula for $\pi_o(x)$ (which was derived from an explicit formula for the Mertens function $M(x)$) at integer values of $x$. Formulas (10) and (18) above are evaluated at an evaluation limit of $J=10$, and the previous formula is evaluated at an evaluation limit of $K=x$ (see formula (1) here). All underlying explicit formulas are evaluated over over the first $200$ pairs of non-trivial zeta zeros. The explicit formulas are illustrated in orange, and the reference function $\pi(x)$ is illustrated in blue.


Illustration of Previous Formula Derived from M(x)

Figure (3): Illustration of Previous Formula for $\pi_o(x)$ Derived from $M_o(x)$


Illustration of Formula (10)

Figure (4): Illustration of Formula (10) for $\pi_o(x)$ Derived from $\psi_o(x)$


Illustration of Formula (18)

Figure (5): Illustration of Formula (18) for $\pi_o(x)$ Derived from $\Pi_o(x)$


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    $\begingroup$ Under the RH the explicit formula for $\sum_{n \le x} \mu(n)$ is proven p.374 , 14.27 of Titchmarsh's book. It is pointwise convergence but the bounds (on the boundary of the integration rectangle to which the residue theorem is applied) let us show local uniform convergence away from the integers and convergence in $L^p_{loc}$. We need to choose a good sequence of $T_v$ (to cross the critical strip) to ensure convergence $\endgroup$ – reuns Apr 25 at 2:08

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