11 Illustration of Derived Formulas Related to Riemann Zeta Function $\zeta[s]$

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There are several relationships between Riemann’s $\zeta[s]$ function and the prime counting functions which are illustrated below. All of these relationships are valid for $\Re[s]>1$. Convergence of relationship (5) below at $\Re[s]=1$ is equivalent to the prime number theorem, and assuming the Riemann hypothesis relationship (5) below actually converges for  $\Re[s]>\frac{1}{2}$.

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

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

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

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

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

The Fourier series for $J[x]$, $J'[x]$, $\psi[x]$, and $\psi'[x]$ have been used to derive formulas for the integrals associated with the five relationships above. Pages 11.1 and 11.2 illustrate the formulas for the integrals related to $\log\zeta[s]$ and $\frac{\zeta'[s]}{\zeta[s]}$ respectively. Here are a few general observations.

  • The method 2 formulas seem to exhibit quicker convergence than the method 1 formulas with respect to the evaluation frequency. The disadvantage of the method 2 formulas is they have longer evaluation times than the method 1 formulas.
  • The formulas for $\log\zeta[s]$ seem to converge quicker than the formulas for $\frac{\zeta'[s]}{\zeta[s]}$ with respect to the upper integration bound. Note that the formulas for $\log\zeta[s]$ are derived from the Fourier series for $J[x]$ and $J'[x]$, whereas the formulas for $\frac{\zeta'[s]}{\zeta[s]}$ are derived from the Fourier series for $\psi[x]$ and $\psi'[x]$. I believe the reason the formulas for $\log\zeta[s]$ seem to exhibit quicker convergence than the formulas for $\frac{\zeta'[s]}{\zeta[s]}$ is perhaps because $J[x]$ is a much slower growing function than $\psi[x]$.
  • The formulas derived from the first-order derivatives seem to exhibit better convergence than the formulas derived from the base prime counting functions which I address further below.

The relationships above imply the following equalities.

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

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

The two equalities above can be shown to be true via integrations by parts. While the two equalities above are true for fully evaluated integrals of fully converged versions of $J'[x]$, $J[x]$, $\psi'[x]$, and $\psi[x]$ for $\Re[s]>1$, they do not hold for partially evaluated integrals of partially converged versions of these functions for $\Re[s]>1$. Integration by parts leads to the following two equalities which are true for partially evaluated integrals of partially converged versions of these functions as well as fully evaluated integrals of fully converged versions of these functions. Evaluation of the left and right sides of the two equalities below leads to two different formulas for $\log\zeta[s]$ and two different formulas for $\frac{\zeta'[s]}{\zeta[s]}$, but since the two different formulas for each function seem to evaluate exactly the same, I haven’t bothered to illustrate the formulas associated with the right side of the two equalities below.

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

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

Pages 11.3 and 11.4 illustrate method 1 formulas for $\zeta[s]$ and $\zeta'[s]$ which were derived from the following relationships.

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

(11) $\quad\zeta'[s]=\int_0^\infty T'[x]\ x^{-s}dx,,\quad\Re[s]>1$