11.2 Illustration of Formulas Derived for $\frac{\zeta'[s]}{\zeta[s]}$ from $\psi[x]$ and $\psi'[x]$

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The following relationships were used to derive formulas for $\frac{\zeta'[s]}{\zeta[s]}$ from the Fourier series representation of the second Chebyshev function $\psi[x]$ and its first-order derivative $\psi'[x]$. Convergence of relationship (3) below for $\Re(s)=1$ is equivalent to the prime number theorem, but assuming the Riemann hypothesis relationship (3) below actually converges for $\Re[s]>\frac{1}{2}$.

(1) $\quad\frac{\zeta'[s]}{\zeta[s]}=-\int_0^\infty\psi'[x]\ x^{-s}\ dx\ ,\quad\Re[s]>1$

(2) $\quad\frac{\zeta'[s]}{\zeta[s]}=-s\int_0^\infty\psi[x]\ x^{-s-1}\ dx\ ,\quad\Re[s]>1$

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


The plots on this page illustrate the method 1 formula derived for $\frac{\zeta'[s]}{\zeta[s]}$ from $\psi'[x]$ via the first relationship above seems to exhibit better convergence than the method 1 formula derived for $\frac{\zeta'[s]}{\zeta[s]}$ from $\psi[x]$ via the second relationship above. I believe this is because the second relationship above is based on a simplification of the following relationship which is derived via integration by parts of the first relationship above, and while the first term on the right side of the following equality (which is ignored in the second relationship above) vanishes for $\Re[s]>1$ as the upper integration bound approaches infinity, it does not vanish for the small upper integration bounds used in the plots on this page.

(4) $\quad\int_0^\infty\psi'[x]\ x^{-s}\ dx=\left.x^{-s}\psi[x]\ \right|_0^\infty+s\int_0^\infty\psi[x]\ x^{-s-1}\ dx\ ,\quad\Re[s]>1$


The following relationships were used to derive additional formulas for $\frac{\zeta'[s]}{\zeta[s]}$ from the Fourier series representation of the Riemann’s prime-power counting function $J[x]$ and its first-order derivative $J'[x]$. These additional formulas are not currently illustrated on this website.

(5) $\quad\frac{\zeta'[s]}{\zeta[s]}=-\int_0^\infty\left(\log[x]\ J'[x]\right)\ x^{-s}\ dx\ ,\quad\Re[s]>1$

(6) $\quad\frac{\zeta'[s]}{\zeta[s]}=-s\int_0^\infty\left(\int_0^x\ \log[x]\ J'[x]\ dx\right)\ x^{-s-1}\ dx\ ,\quad\Re[s]>1$

(7) $\quad\frac{\zeta'[s]}{\zeta[s]}=-s\left(\frac{1}{s-1}+\int_1^\infty\left(\left(\int_0^x\ \log[x]\ J'[x]\ dx\right)-x\right)\ x^{-s-1}\ dx\right),\quad\Re[s]\geq 1$


The dashed red reference curve in the following two plots is $\Re\left[\frac{\zeta'[s]}{\zeta[s]}\right]$ which was evaluated along the line $s=2+i\ t$ using the $Zeta[s]$ and $Zeta'[s]$ functions defined in the Wolfram Language. The remaining curves in the two plots below were generated using the method 1 formulas derived for $\frac{\zeta'[s]}{\zeta[s]}$ from relationships (1) and (2) above with evaluation frequencies $f$=$1$ (orange), $f$=$2$ (green), and $f$=$4$ (blue). All three of these latter curves were evaluated using a lower integration bound of $x$=$1$ and an upper integration bound of $x$=$35$. The lower integration bound $x$=$1$ was used instead of $x$=$0$ because the derived formulas for the indefinite integrals exhibit infinities when evaluated at $x$=$0$. The plots below illustrate for$\Re[s]>1$, the method 1 formulas for $\frac{\zeta'[s]}{\zeta[s]}$ derived from the Fourier series representation of $\psi[x]$ and $\psi'[x]$ converge closer and closer to $\frac{\zeta'[s]}{\zeta[s]}$ (dashed red reference curve) as the evaluation frequency increases.

$Re(\frac{\zeta'(2+iz)}{\zeta(2+iz)})$ derived from $\psi(x)$ evaluated at $f$=$1$ (orange), $f$=$2$ (green), and $f$=$4$ (blue).
$R\left[\frac{\zeta'[s]}{\zeta[s]}\right]$ at $s=2+i\ t$ derived from $\psi[x]$ via relationship (2) evaluated at $f$=$1$ (orange), $f$=$2$ (green), and $f$=$4$ (blue).
$Re(\frac{\zeta'(2+iz)}{\zeta(2+iz)})$ derived from $\psi'(x)$ evaluated at $f$=$1$ (orange), $f$=$2$ (green), and $f$=$4$ (blue).
$\Re\left[\frac{\zeta'[s]}{\zeta[s]}\right]$ at $s=2+i\ t$ derived from $\psi'[x]$ via relationship (1) evaluated at $f$=$1$ (orange), $f$=$2$ (green), and $f$=$4$ (blue).


The following two plots illustrate relationship (3) above converges along the convergence boundary line $s=1+i\ t$ for both the real part (first plot below) and the imaginary part (second plot below). The dashed orange reference functions in the two plots below were generated using the Zeta[s] and Zeta'[s] functions defined in the Wolfram Language. The blue curves in the two plots below were generated using the method 1 formula derived for $\frac{\zeta'[s]}{\zeta[s]}$ from relationship (3) above using a lower integration bound of $x$=$1$, an upper integration bound of $x$=$35$, and an evaluation frequency $f$=$4$. The red discrete portions of the two plots below represent evaluation of the method 1 formula at $s=1+i\,t$ where $t$ corresponds to the the imaginary part of the first ten zeta zeros.

Real part of $\frac{\zeta'(s)}{\zeta(s)}=-s(\frac{1}{s-1}+\int_{1}^{N}x^{-s-1}(\psi(x)-x)dx)$ evaluated at $s=1+i z$, $N$=$35$, and $f$=$4$.
$\Re\left[\frac{\zeta'[s]}{\zeta[s]}\right]$ derived from $\psi[x]$ via relationship (3) evaluated at $s=1+i\ t$, $N$=$35$, and $f$=$4$.
Imaginary part of $\frac{\zeta'(s)}{\zeta(s)}=-s(\frac{1}{s-1}+\int_{1}^{N}x^{-s-1}(\psi(x)-x)dx)$ evaluated at $s=1+i z$, $N$=$35$, and $f$=$4$.
$\Im\left[\frac{\zeta'[s]}{\zeta[s]}\right]$ derived from $\psi[x]$ via relationship (3) evaluated at $s=1+i\ t$, $N$=$35$, and $f$=$4$.


The following two plots illustrates relationship (3) above evaluated along the critical line $s=\frac{1}{2}+i\ t$ diverges at the location of the zeta zeros for both the real part (first plot below) and imaginary part (second plot below). The dashed orange reference functions in the two plots below were generated using the Zeta[s] and Zeta'[s] functions defined in the Wolfram Language. The blue curves in the two plots below were generated using the method 1 formula derived for $\frac{\zeta'[s]}{\zeta[s]}$ from relationship (3) above using a lower integration bound of $x$=$1$, an upper integration bound of $x$=$35$, and an evaluation frequency $f$=$4$. The red discrete portions of the two plots below represent evaluation of the method 1 formula at the first ten zeta zeros.

Real part of $\frac{\zeta'(s)}{\zeta(s)}=-s(\frac{1}{s-1}+\int_{1}^{N}x^{-s-1}(\psi(x)-x)dx)$ evaluated at $s=\frac{1}{2}+i z$, $N$=$35$, and $f$=$4$.
$\Re\left[\frac{\zeta'[s]}{\zeta[s]}\right]$ derived from $\psi[x]$ via relationship (3) evaluated at $s=\frac{1}{2}+i\ t$, $N$=$35$, and $f$=$4$.
Imaginary part of $\frac{\zeta'(s)}{\zeta(s)}=-s(\frac{1}{s-1}+\int_{1}^{N}x^{-s-1}(\psi(x)-x)dx)$ evaluated at $s=\frac{1}{2}+i z$, $N$=$35$, and $f$=$4$.
$\Im\left[\frac{\zeta'[s]}{\zeta[s]}\right]$ derived from $\psi[x]$ via relationship (3) evaluated at $s=\frac{1}{2}+i\ t$, $N$=$35$, and $f$=$4$.