10.3 – Illustration of Formulas Derived for $\log\zeta[s]$ from $J[x]$ and $J'[x]$

The following relationships were used to derive formulas for $\log\zeta[s]$ from the Fourier series representation of Riemann’s prime-power counting function $J[x]$ and the first-order derivative $J'[x]$.

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

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

The plots on this page illustrate the method 1 formula derived for $\log\zeta[s]$ from $J'[x]$ seems to exhibit better convergence than the method 1 formula derived for $\log\zeta[s]$ from $J[x]$. 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.

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

The dashed red reference curve in the following two plots is $\Re[\log\zeta[s]]$ which was evaluated along the line $s=2+i\ t$ using the $Zeta[s]$ function defined in the Wolfram Language. The remaining curves in the two plots below were generated using the method 1 formulas derived for $\log\zeta[s]$ 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 $x$=$11$. 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 $\log\zeta[s]$ derived from the Fourier series representation of $J[x]$ and $J'[x]$ converge closer and closer to $\log\zeta[s]$ (dashed red reference curve) as the evaluation frequency increases.

The two plots below use the same upper integration bound as the two plots above, and illustrate the method 2 formulas derived for $\log\zeta[s]$ seem to exhibit faster convergence with respect to the evaluation frequency limit than the method 1 formulas. The convergence of integrals associated with the method 2 formulas is a bit tricky as they seem to be more sensitive to the lower integration bound, and the lower integration bound used here for the second plot below works well for $f>1$, but not very well for $f=1$.

The plots above were for $\Re[s]=2$ since the relationships above are valid for $\Re[s]>1$, but the following plots explore the convergence of the derived formula for $\log\zeta[s]$ from $J'[x]$ via relationship (1) for $\Re[s]\in \{\frac{1}{2},\ 1\}$. In general, convergence seems to increase for $\Re[s]\in \{\frac{1}{2},\ 1\}$ as the evaluation frequency and imaginary part of $s$ increases. There is an oscillation in the evaluations of the derived formula for $\log\zeta[s]$ below which is most noticeable for low magnitudes of the imaginary part of $s$. The frequency and amplitude of this oscillation both increase as the upper integration bound increases, but the amplitude decreases as the imaginary portion of $s$ increases.

The dashed orange reference curves in the following two plots are $\Re[\log\zeta[s]]$ and $\Im[\log\zeta[s]]$ which were evaluated along the convergence boundary line $s=1+i\ t$ using the $Zeta[s]$ function defined in the Wolfram Language. The blue curves in the plots were generated using the method 1 formula derived for $\log\zeta[s]$ from $J'[x]$ using a lower integration bound of $x$=$1$, an upper integration bound of $x$=$32$, and an evaluation frequency $f$=$4$. These plots illustrate convergence seems to increase as the magnitude of t increases when the formula derived for $\log\zeta[s]$ is evaluated along the convergence boundary line $s=1+i\ t$.

The dashed orange reference curves in the following two plots are $\Re[\log\zeta[s]]$ and $\Im[\log\zeta[s]]$ which were evaluated along the critical line $s=\frac{1}{2}+i\ t$ using the $Zeta[s]$ function defined in the Wolfram Language. The blue curves in the plots were generated using the method 1 formula derived for $\log\zeta[s]$ from $J'[x]$ via relationship (1) using a lower integration bound of $x$=$1$, an upper integration bound of $x$=$32$, and an evaluation frequency $f$=$4$. These plots illustrate convergence seems to increase as the magnitude of t increases when the formula derived for $\log\zeta[s]$ is evaluated along the critical line $s=\frac{1}{2}+i\ t$.