Convergence of Real Part of $\int_{1-\epsilon}^\infty \left(\left(\sum_n \delta[x-n]\right)-1\right)\,x^{-s}\,dx$ for $Re[s]\in(0,\,1)$

Assuming the following definition for $G[s]$, the context of this post is the convergence of $Re[G[s]]$ to $Re[\zeta[s]]$ for $Re[s]\in(0,\,1)$.

(1) $\quad G[s]=\int_{1-\epsilon}^N\left(\left(\sum_n \delta[x-n]\right)-1\right)\,x^{-s}\,dx\,,\ N\to\infty$

I’ve noticed $Re[G[s]]$ seems to approximate $Re[\zeta[s]]$ very closely for $Re[s]=1$ as $N\to\infty$.

The following two plots illustrate $Re[G[1+i\,t]]-Re[\zeta[1+i\,t]]$ for $N=100$ and $N=1000$ respectively. Note the amplitude of the error oscillation is virtually the same across the entire range of $t$ for both of the plots, and as the value of $N$ increases by an order of magnitude from the first to the second plot, the amplitude of the error oscillation seems to decrease by an order of magnitude from the first to the second plot.

$Re[G[1+i\,t]]-Re[\zeta[1+i\,t]]$ for $N=100$
$Re[G[1+i\,t]]-Re[\zeta[1+i\,t]]$ for $N=1000$
 I’ve been wondering if the Prime Number Theorem predicts $Re[G[s]]$ converges to $Re[\zeta[s]]$ for $Re[s]=1$ as $N\to\infty$.

I’ve also noticed $Re[G[s]]$ seems to approximate $Re[\zeta[s]]$ with an error bound of $\frac{1}{2}$ for $Re[s]=0$ as $N\to\infty$ and $Im(s)\to\infty$.

The following two plots illustrate $Re[G[i\,t]]-Re[\zeta[i\,t]]$ for $N=100$ and $N=1000$ respectively. Note that as the value of $N$ increases by an order of magnitude from the first to the second plot, there is no discernible decrease in the amplitude of the error oscillation from the first to the second plot.

$Re[G[i\,t]]-Re[\zeta[i\,t]]$ for $N=100$
$Re[G[i\,t]]-Re[\zeta[i\,t]]$ for $N=1000$
I’ve been wondering if the Riemann Hypothesis predicts $Re[G[s]]$ approximates $Re[\zeta[s]]$ with an error bound of $\frac{1}{2}$ for $Re[s]=0$ as $N\to\infty$ and $Im(s)\to\infty$.

The formula I’m using to evaluate $G[s]$ is provided in (2) below to aid others who might be interested in exploring this relationship for themselves. All four plots above use the value $\epsilon=0.000001$ and the $Zeta[s]$ function provided by the Wolfram language as the reference for $\zeta[s]$.

(2) $\quad G(s)=\frac{\left((\epsilon -1) N^s+N (1-\epsilon )^s\right) (N-N \epsilon )^{-s}}{s-1}+\sum _{n=1}^N n^{-s}\ ,\ N\to\infty$

Convergence of $\int_1^N (\psi[x]-x)\ x^{-s-1}\ dx$ with Respect to Evidence of the Riemann Hypothesis

The context of this post is convergence of the following integral.

(1)  $\int_1^N\left(\psi[x]-x\right)\ x^{-s-1}\ dx=-\frac{\zeta'[s]}{s\ \zeta[s]}+\frac{1}{1-s}\ ,\ N\to\infty$

I’ve been told the prime number theorem implies integral (1) above converges absolutely for $\Re[s]=1$, and the convergence of this integral for some given $s$, $\Re[s]\in\left(\frac{1}{2},\ 1\right)$, is equivalent to the lack of zeros of $\zeta[z]$ on $\Re[z]\ge \Re[s]$. I’ve noticed this integral seems to converge for $\Re[s]=\frac{1}{2}+\epsilon$ which is consistent with the Riemann hypothesis.

I initially explored integral (1) above a few months ago using a few of my formulas for $\psi_0[x]$, and some of the results are illustrated on page 11.2  At that time I explored convergence along the line $s=1+i\ t$ and along the critical line $s=\frac{1}{2}+i\ t$. More recently I’ve explored convergence of some of my formulas along the line $s=\left(\frac{1}{2}+\epsilon\right)+i\ t$ and some of the results are illustrated later below. I’ve also explored integral (1) above using two additional representations of $\psi[x]$ including firstly von Mangoldt’s explicit formula for $\psi_0[x]$ and secondly in the context of distributions and Dirac delta functions, and some of the results from the second additional approach are also illustrated further below.

Note that von Mangoldt’s formula for $\psi_0[x]$ has an infinity at $x=1$. I’ve gotten around this infinity by using a series expansion (evaluated with finite limits) for the associated small error term in order to evaluate integral (1) above, but I’m somewhat skeptical that this is a valid approach.

My formulas for $\psi_0[x]$ all converge for $x>0$, and with suitable selection of evaluation parameters some of them converge at $x=0$ (in which case they evaluate to perfect odd functions). So perhaps my formulas have an advantage over von Mangoldt’s formula with respect to the evaluation of integral (1) above at the lower integration bound $x=1$. I also believe the convergence of my formulas for $\psi_0[x]$ for $x>0$ can be rigorously proven as the underlying mathematics are relatively elementary and fundamental.

I’ve been told assuming the Riemann Hypothesis integral (1) above converges as $N\to\infty$ only for $\Re[s]>\frac{1}{2}$, and hence you can approximate the zeros of $\zeta[s]$ on $\Re[s]=\frac{1}{2}$ by knowledge of $\psi[x]$ for $x<N$ if and only if the Riemann hypothesis is true.

The three plots below illustrate the real, imaginary, and absolute value of integral (1) above. The blue curves in the three plots below illustrate the left side of integral (1) above evaluated with one of my formulas for $\psi_0[x]$ along the vertical line $\Re[s]=\frac{1}{2}+\epsilon$ where $\epsilon=0.01$. The evaluations were performed using a lower integration bound of 1, an upper integration bound of $N=154$, and an evaluation frequency $f=4$. The orange reference curves in the three plots below illustrate the right side of integral (1) above evaluated using the $Zeta[s]$ and $Zeta'[s]$ functions provided in the Wolfram language. The red discrete points in the three plots below illustrate the evaluation of the left side of integral (1) above at $s=ZetaZero[i]+\epsilon$ for the first 10 zeta zeros.

The three plots below illustrate the left side of integral (1) evaluated with my formulas for $\psi_0[x]$ seems to converge to the reference function on the right side of integral (1) for $Re[s]=\frac{1}{2}+\epsilon$ where $\epsilon=0.01$. The high-frequency oscillation visible in the plot seems to increase in frequency and decrease in magnitude as evaluation limits $N$ and $f$ increase. In order to reduce evaluation time, I specified $PlotPoints->100$ and $MaxRecursion->0$ when generating the three plots below, and it’s likely the high-frequency oscillation would appear a bit less ragged at higher plot resolution. Another undesirable effect of the limited plot resolution used in the following three plots is the vertical spires of the orange reference function have been filtered out in some cases.

The three plots below seem to illustrate the location of the zeros of $\zeta[s]$ on $\Re[s]=\frac{1}{2}$ can be approximated by knowledge of $\psi[x]$ for $x<N$, and this seems to imply that the Riemann hypothesis is true.

Real Part of Integral (1)
Imaginary Part of Integral (1)
Absolute Value of Integral (1)

The convergence of integral (1) above can be explored in a much simpler context than von Mangoldt’s formula or my formulas for $\psi_0[x]$. I derived the following formula for the evaluation of integral (1) via integration by parts using the context of distributions and Dirac delta functions.

(2)  $\int_1^N\left(\psi[x]-x\right)\ x^{-s-1}\,dx=\frac{N^{1-s}-1}{s-1}+\frac{1}{s}\sum _{n=1}^N \Lambda [n] \left(n^{-s}-N^{-s}\right),\\$
$\quad \quad N\to\infty$

The three plots below which use formula (2) to evaluate integral (1) above illustrate similar results to the three plots above which used one of my formulas for $\psi_0[x]$ to evaluate integral (1) above. I used the same value for $\epsilon$ (0.01) but a higher upper integration limit $N=500$ since formula (2) evaluates much faster than the corresponding formula for integral (1) derived from my formula for $\psi_0[x]$. The plots below use default values for the $PlotPoints$ and $MaxRecursion$ options for the same reason. The higher resolution of the following three plots results in a more accurate representation of the vertical spires in the orange reference function than the previous three plots which used a lower resolution.

Again, the three plots below seem to illustrate the location of the zeros of $\zeta[s]$ on $\Re[s]=\frac{1}{2}$ can be approximated by knowledge of $\psi[x]$ for $x<N$, and again, this seems to imply that the Riemann hypothesis is true.

Real Part of Integral (2)
Imaginary Part of Integral (2)
Absolute Value of Integral (2)