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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

$\pi(x) - \text{Li}(x)$?

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1  
I fail to understand what your saying at the point in which you introduce the function $c(x)$, I recommend formatting your text properly. – Ethan 2 days ago
I don't understand what you mean by " do some fourier analysis on the function $\pi(x)-\frac{x}{\ln(x)}$ ". If you are asking for a good estimate on the difference between $\pi(x)$ and $\frac{x}{\ln(x)}$, then this would be asymptotic to $\text{Li}(x)-\frac{x}{\ln(x)}$, this is because the logarithmic integral is a much better approximation to $\pi(x)$ then $\frac{x}{\ln(x)}$, so with that said it would probably be more interesting to study the behavior of $\pi(x)-\text{Li(x)}$. – Ethan 2 days ago
basically, I want to see if there are any patterns or noticeable trends in the difference, that can be approximated by sines – frogeyedpeas 2 days ago
The difference between $\pi(x)$ and $\frac{x}{\ln(x)}$ can already be approximated very well, I think the question to ask would be regarding the difference of $\pi(x)$ and $\text{Li}(x)$. – Ethan 2 days ago
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The explicit formula for the prime counting function is perhaps the 'correct' mixture of fourier analysis (using the zeros of the Riemann zeta function) and asymptotics (employing the logarithmic integral function). The PNT $\pi(x)\sim x/\log x$ is equivalent to $\psi(x)\sim x$, where $\psi(x)$ is the chebyshev function, a variant way of measuring prime growth. A more direct fourier expansion of $\psi(x)-x$ also exists, a variant of the explicit formula. See the link for more details. – anon 2 days ago

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