The inspiration for the approach illustrated in this post is the book “Prime Numbers and the Riemann Hypothesis” by Barry Mazur and William Stein.

The first two plots below illustrate $\psi'[e^{Abs[t]}]-1$ (where $\psi'[x]$ is the Fourier series for the first-order derivative of the second Chebyshev function) at a frequency evaluation limit of $f=1$. The orange reference function in these two plots is $2\ f\log\ e^{Abs[t]}$.

The next two plots below illustrate $e^{-Abs[t/2]}\left(\psi'[e^{Abs[t]}]-1\right)$ (where $(\psi'[x]$ is the Fourier series for the first-order derivative of the second Chebyshev function) at a frequency evaluation limit of $f=1$. The orange reference function in these two plots is $2\ f\ e^{-Abs[t/2]}\log\ e^{Abs[t]}$.

The last plot below illustrates the evolution of the zeta zeros from the real portion of the Fourier transform of $e^{-t/2}\left(\psi'[e^t]-1\right)$ (where $\psi'[x]$ is the Fourier series for the first-order derivative of the second Chebyshev function) at a frequency evaluation limit of $f=1$. The Fourier transform is evaluated along the critical line (i.e. $Re[s]=\frac{1}{2}$).

The context of this post is the relationship between evaluation of the partial integral for $\psi(x)$ defined in (1) below and partial evaluation of von Mandgoldt’s explicit formula for $\psi(x)$ defined in (2) below.

I’ve been wondering if it’s true that for some value of $b_m$ where $Im[ZetaZero[m]]<b_m<Im[ZetaZero[m+1]]$, evaluation of integral (1) above is equivalent to evaluating von Mangoldt’s formula (2) above over the first $m$ conjugate pairs of zeta zeros, and if so whether theres a way to determine the intermediate value of $b_m$ other than using successive approximation.

The plots below illustrate evaluation of the partial integral (1) above approximates partial evaluation of von Mandoldt’s explicit formula (2) above through the 10th and 11th zeta zeros, and the difference between these two approximates the contribution of the 11th zeta zero in von Mandoldt’s explicit formula (2) above. I’m not sure how the two small error terms in von Mandoldt’s formula evolve over the range of integral (1) above, but I suspect they evolve fairly quickly, and if this is the case it seems evaluation of integral (1) above through $b_m$ will approximate partial evaluation of von Mandgoldt’s explicit formula (2) above through zeta zero $m$ with increasing accuracy as $m$ increases.

The orange curves in the following two plots illustrate partial evaluation of von Mangoldt’s explicit formula (2) above through the 10th and 11th zeta zeros respectively. Note there is a slight increase in sharpness of the second plot below since it includes evaluation of an additional zeta zero. The blue curves in the two plots below illustrate evaluation of integral (1) above using integration bounds corresponding to an estimate of $b_{10}$ in the first plot below and an estimate of $b_{11}$ in the second plot below, but the evaluations of (1) and (2) are so similar that at this scale the evaluations of (1) in blue are mostly hidden by the evaluations of (2) in orange, and therefore the green curves were added to the following two plots to illustrate the difference between the evaluations of (1) and (2).

The orange curve in the following plot illustrates evaluation of the contribution of the 11th zeta zero in von Mangoldt’s formula (2) above. The blue curve in the following plot illustrates evaluation of formula (3) below using estimates of $b_{11}$ and $b_{10}$ in the first and second integrals of formula (3) respectively, but the evaluations of (2) and (3) are so similar that at this scale the evaluation of (3) in blue is mostly hidden by the evaluation of (2) in orange, and therefore the green curve was added to the following plot to illustrate the difference between the evaluations of (3) and (2).

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.

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.

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.