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This question assumes the following definitions.

(1) $\quad\psi(x)=\sum\limits_{n\le x}\Lambda(n)\qquad\text{(second Chebyshev function)}$

(2) $\quad\Pi(x)=\sum\limits_{n\le x}\frac{\Lambda(n)}{\log(n)}\qquad\text{(Riemann's prime-power counting function)}$

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

(4) $\quad\widetilde{\Pi}(x)=\sum\limits_{n\le x}\frac{\Lambda(n+1)}{\log(n+1)}=\Pi(x+1)$


My primary motivation for investigating $\widetilde{\psi}(x)$ and $\widetilde{\Pi}(x)$ defined in (3) and (4) above is a Dirichlet inverse can be derived for $\Lambda(n+1)$ and $\frac{\Lambda(n+1)}{\log(n+1)}$ but not for $\Lambda(n)$ and $\frac{\Lambda(n)}{\log(n)}$. This allows a wide variety of functions to be expressed in terms of $\psi(x)$ and $\Pi(x)$, and consequently von Mangoldt's explicit formula for $\psi(x)$ and Riemann's explicit formula for $\Pi(x)$ can be used to derive explicit formulas for a wide variety of functions.


For example consider the divisor summatory function defined in (5) below.

(5) $\quad D(x)=\sum\limits_{n\le x}\sigma_0(n)\qquad\text{(divisor summatory function)}$


The following plot illustrates an explicit formula for $D(x)$ derived from the explicit formula for $\psi(x)$ where the evaluation is over the first 200 pairs of non-trivial zeta zeros. The explicit formula for $D(x)$ is illustrated in orange and the corresponding reference function is illustrated in blue. The red discrete portion of the plot illustrates evaluation of the explicit formula for $D(x)$ at integer values of $x$. The dashed-green reference curve illustrates the asymptotic function $(2\,\gamma-1)\,n+n\,\log(n)$.

Illustration of Explicit Formula for $D(x)$ Derived from Explicit Formula for $\psi(x)$

$\text{Figure (1): Illustration of Explicit Formula for $D(x)$ Derived from Explicit Formula for $\psi(x)$}$


The Dirichlet transforms of $\Lambda(n)$ and $\frac{\Lambda(n)}{\log(n)}$ are as follows.

(6) $\quad\sum\limits_{n=1}^\infty\frac{\Lambda(n)}{n^s}=-\frac{\zeta'(s)}{\zeta(s)}\,,\qquad\quad\Re(s)>1$

(7) $\quad\sum\limits_{n=1}^\infty\frac{\Lambda(n)}{\log(n)\,n^s}=\log\zeta(s)\,,\quad\Re(s)>1$


I'm wondering about the corresponding Dirichlet transforms of $\Lambda(n+1)$ and $\frac{\Lambda(n+1)}{\log(n+1)}$ defined in (8) and (9) below.

(8) $\quad\sum\limits_{n=1}^\infty\frac{\Lambda(n+1)}{n^s}=?$

(9) $\quad\sum\limits_{n=1}^\infty\frac{\Lambda(n+1)}{\log(n+1)\,n^s}=?$


Question (1): What is the Dirichlet transform of $\Lambda(n+1)$ defined in (8) above? What are the locations of the trivial and non-trivial poles and zeros of this Dirichlet transform?

Question (2): What is the Dirichlet transform of $\frac{\Lambda(n+1)}{\log(n+1)}$ defined in (9) above? What are the locations of the trivial and non-trivial poles and zeros of this Dirichlet transform?

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