Fourier Analysis of Prime Counting Function I was thinking about the following:
Denote $\pi(x)$ as the prime counting function such that:
$$
\pi(x) = \#\text{ of prime numbers}\leq x
$$
It is well known from the prime number theorem that
$$
\pi(x) \sim \frac{x}{\ln x}
$$
and
$$
\pi(x) \sim \text{Li}(x),\quad \text{Li}(x)=\int_2^x\frac{1}{\ln t}\,dt.
$$
Note the following:
if $A(x) \sim B(x)\implies A(x)/B(x) = 1 \text{ as }x\to\infty$ and $A(x), B(x)\to\infty \text{ as }x\to\infty$.
If $C(x)$ is a function such that $n \leq C(x) \leq k$ then:
$$(A(x) + n)/B(x) \leq (A(x) + C(x))/B(x) \leq (A(x) + k)/B(x)$$ 
$$A(x)/B(x) + n/B(x) \leq A(x)/B(x) + C(x)/B(x) \leq A(x)/B(x) + k/B(x)$$
If x is taken to infinity:
$$1 + 0 \leq A(x)/B(x) + C(x)/B(x) \leq 1 + 0$$
$$\rightarrow A(x) + C(x) ~ B(x)$$
What interests me is that since we already know that:
$$\pi(x) \sim x/\ln(x)$$
and from above that $x/ln(x) +$ any number of functions of the form $A(b(x)) ~ \pi(x)$
Can we not try to do some sort of fourier analysis on the function:
$$\pi(x) - x/\ln(x) $$
or 
$\pi(x) - \text{Li}(x)$?
 A: The purpose of $\zeta(s)$ is to perform Fourier analysis on $\pi(e^u)$ :
$$\zeta(s) = \prod_p \frac{1}{1-p^{-s}} \quad \implies \quad \ln \zeta(s) = \sum_{p^k} \frac{p^{-sk}}{k}$$
i.e. it is the $\color{red}{\text{Laplace transform}}$ of $\sum_{p^k} \frac{1}{k}\delta(u-\ln p^k)$ $\qquad \scriptstyle \text{(or if you prefer, }\hat{f}(\xi ) =\ln \zeta(\sigma+2i \pi \xi) \text{ is the Fourier transform of } f(u) = \sum_{p^k} \frac{1}{\scriptstyle k \ p^{ \sigma k}}\delta(u-\ln p^k)$)
Integrating by parts, you get
$$\ln \zeta(s) = s \int_0^\infty J(e^u) e^{-su}du$$
where $J(x) = \sum_{p^k < x}\frac{1}{k} = \pi(x) + \mathcal{O}(\sqrt{x})$.
i.e. 
$$\boxed{\frac{\ln \zeta(s) }{s} = h(s) + \int_0^\infty \pi(e^u) e^{-su}du}$$
where $h(s) = \int_0^\infty (J(e^u)-\pi(e^u))e^{-su}du$ is analytic and bounded for $Re(s) > 1/2+\epsilon$.
A: You can perform Fourier analysis on $\pi(x)$. Here are two plots of a Fourier series for $\pi'(x)$, i.e. the derivative of $\pi(x)$, with a frequency evaluation limit $f=1$ for the first plot and $f=4$ for the second plot. The two plots below illustrate the general relationship that this Fourier series always evaluates to $2f$ times the step size of $\pi(x)$ at integer values of $x$.
Fourier Series for π'(x) Evaluated at f=1
Fourier Series for π'(x) Evaluated at f=4
