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

$\vartheta”[x]=\frac{d}{dx}(\log[x]\ \pi'[x])\\$

$\psi”[x]=\frac{d}{dx}(\log[x]\ J'[x])\\$

$M”[x]=\frac{d}{dx}(log[x]\ K'[x])$

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

$\pi”[x]=\frac{d}{dx}(\frac{\vartheta'[x]}{\log[x]})\\$

$J”[x]=\frac{d}{dx}(\frac{\psi'[x]}{\log[x]})\\$

$K”[x]=\frac{d}{dx}(\frac{M'[x]}{\log[x]})$

In summary, I’ve derived a total of 36 formulas for second-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 6.4-6.9 below currently focus on the initial formulas derived via method 1. I’m considering perhaps adding illustrations of all six formulas for $J”[x]$ and all six formulas for $\psi”[x]$ sometime in the future.

In general, the Fourier series for the second-order derivatives of prime counting functions always evaluate to zero at $x=0$ and are always odd functions of x (i.e. $f”[-x]=-f”[x]$). There are three additional items noticeable in the illustrations of the Fourier series for the second-order derivatives of the prime counting functions.

- First, the evaluation is not exactly zero at primes (or prime powers), which is consistent with the evaluation at primes (or prime powers) not being at the peak of their associated primary lobe in the illustrations of the Fourier series for the first-order derivatives of the prime counting functions.
- Second, there is a significant positive spike on the left side of each prime (or prime power) and a significant negative spike on the right side of each prime (or prime power) which is consistent with a significant primary lobe around each prime (or prime power) in the illustrations of the Fourier series for the first-order derivatives of the prime counting functions.
- Third, there seems to be an oscillation which is noticeable in the discrete evaluations at integers which I believe is related to a strong influence of the first zeta zero.

Pages 6.2 and 6.3 below illustrate plots for the second-order derivatives $S”[x]$ and $T”[x]$ of the rational and log-step stairway functions.

Page 6.1 below illustrates slightly different plots for the second-order derivative $U”[x]$ of the unit step function which are explained on that page.