10.2 Illustration of Formula Derived for $\zeta'[s]$ from $S'[x]$

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The following relationships were used to derive six formulas for $\zeta'[s]$ (i.e. two formulas from each of the three methods for derivation of Fourier series). The second relationship represents integration by parts of the first relationship.

$\zeta'[s]=\frac{d}{ds}(\int_{1-\epsilon}^{\infty}S'[x]\ x^{-s}dx),\quad Re[s]>1\\$
$\zeta'[s]=\frac{d}{ds}(x^{-s}S[x]+s\int_{1-\epsilon}^{\infty}S[x]\ x^{-s-1}dx),\quad Re[s]>1$

Since x and s are independent variables, the order of integration and differentiation can be reversed in the relationships above. The following relationships were used to derive six additional formulas for $\zeta'[s]$ (i.e. two formulas from each of the three methods for derivation of Fourier series).

$\zeta'[s]=\int_{1-\epsilon}^{\infty}S'[x]\ \frac{d\ (x^{-s})}{ds}\ dx,\quad Re[s]>1\\$
$\zeta'[s]=\frac{d\ (x^{-s})}{ds}S[x]+\int_{1-\epsilon}^{\infty}S[x]\ \frac{d\ (s\ x^{-s-1})}{ds}\ dx,\quad Re[s]>1$

The four formulas for each method seem to evaluate exactly the same as each other, so only the formula derived from the first relationship is illustrated in the plots on this page and it’s children pages. This page and it’s children pages currently only illustrates the method 1 formula derived for $\zeta'[s]$ from the first relationship above. All plots on this page and it’s children pages currently use a lower integration bound of $x=\frac{1}{2}$ and an upper integration bound of $50$. The lower integration bound of $x=\frac{1}{2}$ is used instead of $x=0$ because the indefinite integrals exhibit infinities when evaluated at $x=0$.


The following plot illustrates the absolute value of the method 1 formula derived for $\zeta'[s]$ evaluated along the line $s=2+i\ t$ (orange). The Zeta[s] function defined in the Wolfram Language is shown as a reference (blue). This plot illustrates the method 1 formula derived for $\zeta'[s]$ converges for $Re[s]>1$. Note there is a small oscillation in the evaluation of the formula for $\zeta'[s]$ below which corresponds to a small oscillation in the evaluation of the formula for $\zeta[s]$ illustrated on page 10.1 which was somewhat less noticeable. As the upper integration bound increases, the frequency of these oscillations increases and the magnitude of these oscillations decreases for $Re[s]>1$.

Method 1 formula for $Abs[\zeta'[2+i t]]$ (orange).
Method 1 formula for $Abs[\zeta'[2+i t]]$ (orange).

The plot above was for $Re[s]=2$ since the relationships above are valid for $Re[s]>1$, but the children pages below explore the convergence of the method 1 formula derived for $\zeta'[s]$ from the first relationship above along the convergence boundary line $s=1+i\ t$ and along the critical line $s=\frac{1}{2}+i\ t$.

In the plots for $Re[s]=1$ and $Re[s]=\frac{1}{2}$ on the children pages below, the formula for $\zeta'[s]$ exhibits an undesirable oscillation. I’ve isolated the source of this isolation by splitting the $\zeta'[s]$ function into two parts which I refer to as $\zeta_a'[s]$ which is the source of the undesirable oscillation and $\zeta_b'[s]$ which is the remainder of $\zeta'[s]$. As the upper integration bound is increased, the oscillation related to $\zeta_a'[s]$ seems to increase in both frequency and magnitude, and $\zeta_b'[s]$ seems to converge closer and closer to $\zeta'[s]$. As the magnitude of $Im[s]=t$ is increased, the magnitude of the oscillation seems to decrease and the convergence of $\zeta_b'[s]$ to $\zeta'[s]$ seems to increase.

$Re[\zeta'[s]]=Re[\zeta_a'[s]+\zeta_b'[s]]=Re[\zeta_a'[s]]+Re[\zeta_b'[s]]\\$
$Im[\zeta'[s]]=Im[\zeta_a'[s]+\zeta_b'[s]]=Im[\zeta_a'[s]]+Im[\zeta_b'[s]]\\$
$Abs[\zeta'[s]]=Abs[\zeta_a'[s]+\zeta_b'[s]]\neq Abs[\zeta_a'[s]]+Abs[\zeta_b'[s]]$