# 5 Illustrations of Fourier Series Representations of First-Order Derivatives

The formulas for $F[x]\in\{U[x],\ S[x],\ T[x],\ \pi[x],\ J[x],\ K[x],\ \vartheta[x],\ \psi[x],\ M[x]\}$ consist of a linear term which I refer to as $F_a[x]$ and a Fourier series which I refer to as $F_b[x]$ where $F[x]=F_a[x]+F_b[x]$ . The first-order derivatives illustrated on the children pages below only include the first-order derivative $F_b'[x]$ of $F_b[x]$ versus the first-order derivative $F'[x]$ of the entire function $F[x]$.

I’ve derived Fourier series for first-order derivatives of prime counting functions via three different methods resulting in three formulas for each of the six prime counting functions.

I’ve derived three additional formulas for first-order derivatives of each of the three log-step prime counting functions $\vartheta[x]$, $\psi[x]$ and $M[x]$ via the following relationships.

$\vartheta'[x]=\log[x]\ \pi'[x]\\$
$\psi'[x]=\log[x]\ J'[x]\\$
$M'[x]=log[x]\ K'[x]$

I’ve derived three additional formulas for first-order derivatives of each of the three rational-step prime counting functions $\pi[x]$, $J[x]$ and $K[x]$ via the following relationships.

$\pi'[x]=\frac{\vartheta'[x]}{\log[x]}\\$
$J'[x]=\frac{\psi'[x]}{\log[x]}\\$
$K'[x]=\frac{M'[x]}{\log[x]}$

In summary, I’ve derived a total of 36 formulas for first-order derivatives of prime counting functions as follows.

$\pi'[x]$ – 6 formulas
$J'[x]$ – 6 formulas
$K'[x]$ – 6 formulas
$\vartheta'[x]$ – 6 formulas
$\psi'[x]$ – 6 formulas
$M'[x]$ – 6 formulas

Since it would be difficult to illustrate all of these formulas each at multiple evaluation frequency limits, pages 5.4-5.9 below focus on the initial formulas derived via method 1. Each of the pages 5.4-5.9 illustrates five plots of the Fourier series for the first-order derivative $F_b'[x]$ of the associated prime counting function. Each of these five plots is described further below.

• The first plot illustrates the Fourier series for the first-order derivative $F_b'[x]$ of each of the prime counting functions is an even function.
• The second, third, and fourth plots focus on the right-half plane with evaluation frequencies $f=1$, $f=2$, and $f=3$ respectively. These plots illustrate the evaluation of the Fourier series for the first-order derivative $F_b'[x]$ of a prime counting function at a positive integer value of $x$ always equals $2\ f$ times the step size of the associated prime counting function at the positive integer $x$.
• The fifth plot focuses around the evaluation at $x=19$ and includes multiple evaluation frequencies ($f={2, 4, 8}$). This plot illustrates that as the evaluation frequency limit $f$ increases towards infinity, the primary lobe associated with the evaluation at a prime integer (and in some cases a prime-power integer) continues to get narrower and taller  consistent with the notion of a Dirac delta impulse function, while always continuing to evaluate to $2f$ times the step size of the associated prime counting function at the prime integer (or in some cases prime-power integer).

Pages 5.2 and 5.3 below illustrate plots for $S_b'[x]$ and $T_b'[x]$ similar to the plots on pages 5.4-5.9 described above.

Page 5.1 below illustrates slightly different plots for $U_b'[x]$ which are explained on that page.

Page 5.10 below illustrates all 6 formulas for $J_b'[x]$, and page 5.11 below illustrates all 6 formulas for $\psi_b'[x]$. All plots on pages 5.10 and 5.11 use an evaluation frequency limit $f=1$.

In general, the Fourier series for the first-order derivative $F_b'[x]$ of a prime-number counting functions are always even functions (i.e. $F_b'[-x]=F_b'[x]$) and always evaluate to exactly $2\ f$ times the step size of the associated prime-number counting function at positive integer values of $x$ where $f$ is the evaluation frequency limit and assumed to be a positive integer.

There are a couple of additional items that are noticeable in the illustrations of the Fourier series for the first-order derivative $F_b'[x]$ of a prime counting functions, and these two items are related to each other. First, the evaluation at $x=0$ does not always equal zero. Second, the evaluation at a prime (or prime-power) is not exactly at the peak of the primary lobe associated with the prime (or prime-power). This is most noticeable at small primes (or small prime-powers) and at the evaluation frequency $f=1$. This topic is discussed further on page 9.