I've defined three general methods for derivation of formulas for prime counting functions where each prime counting function is represented by an infinite series of Fourier series.
In the distributional framework for prime counting functions the first-order derivatives are represented by Dirac delta distributions. In the case of $\psi'(x)$, there is a Dirac delta distribution at each prime-power value of $x$ with weight $Log(p)$ where $x$ is of the form $x=p^n$. Note the real portion of the Fourier transform of a Dirac delta function is a Cosine term.
(1) $\quad \operatorname{FourierTransform}(\delta(x-a),x,y)=e^{-2\,i\,\pi\,a\,y}=\cos(2\,\pi\,a\,y)-i\,\sin(2\,\pi\,a\,y)\,,\quad a\in\mathbb{R}$
In the Fourier series framework for prime counting functions, the first-order derivatives are represented by infinite series of Fourier series which converge to the Dirac delta distributions in the distributional framework. These Fourier series consist of Cosine terms, and note the Fourier transform of a Cosine function is a pair of Dirac delta distributions.
(2) $\quad \operatorname{FourierTransform}(\cos(2\,\pi\,b\,x),x,y)=\dfrac{\delta(b-y)}{2}+\dfrac{\delta(b+y)}{2}$
The Fourier transforms in (1) and (2) above both assume the Fourier parameters $\{0,\,-2\,\pi\}$.
Question 1: What is the relationship between the Cosine terms in the distributional framework and the Dirac delta distributions in the Fourier series framework? For example, do the Cosine terms in the distributional framework converge to the Dirac delta distributions in the Fourier series framework, analogous to the way the Cosine terms in the Fourier series framework converge to the Dirac delta distributions in the distributional framework?
Question 2: If the analogous relationship in question 2 is valid, is this convergence in any way sensitive to the Fourier parameters used for the two Fourier transforms? For example, will the convergence only apply when using the same set or a specific set of Fourier parameters for both Fourier transforms? Or will the convergence perhaps be faster if the same Fourier parameters are used for both Fourier transforms versus using different Fourier parameters for the two Fourier transforms?
1/1/2018 Update: I believe the discrepancies between the Fourier transforms of the distributional and Fourier series representations of prime-counting functions result from the idealization of the Fourier transform of $sin$ and $cos$ functions as Dirac delta ($\delta$) functions. Please see the answer I posted below which I believe provides a fair amount of insight into the theory and value of Fourier series representations of non-periodic functions. Happy New Year!
Direct link to answer I posted below
5/21/2022 Update:
Formulas (7) and (8) of my question related to the nested Fourier series representation of $h(s)=\frac{i s}{s^2-1}$ define a general method for derivation of a nested Fourier series representation for $f(x)=\sum\limits_{n=1}^x a(n)$ which I believe converges for $x>0$ when the related Dirichlet series $F(s)=\sum\limits_{n=1}^\infty\frac{a(n)}{n^s}$ converges for $\Re(s)\ge 2$.
In some cases, such as the case when $a(n)=\delta_{n,1}$ (Kronecker delta function) which is addressed in my answer below, an evaluation limit can be chosen such that the nested Fourier series for $f(x)$ evaluates to zero at $x=0$ in which case it becomes an odd function of $x$.
This answer I posted to a question about an entire function interpolating $\mu(n)$ defines an alternate analytic representation for $f(x)=\sum\limits_{n=1}^x a(n)$ which I believe more generally converges to zero at $x=0$ (and hence an odd function of $x$) assuming the related Dirichlet series $F(s)=\sum\limits_{n=1}^\infty\frac{a(n)}{n^s}$ converges for $\Re(s)\ge 2$.
Best Answer
Your question means nothing. Work on $$\text{I}\Pi(x) = \sum_{k=-\infty}^\infty \delta(x-k) = 1+2 \sum_{n=1}^\infty \cos(2\pi n x)$$ where the Fourier series on the right converges only is in the sense of distributions, that is, for every $\varphi \in C_c^\infty$ with (*) compact support $[a,b]$ : $$\langle \text{I}\Pi, \varphi \rangle = \int_{-\infty}^\infty \text{I}\Pi(x) \varphi(x) \, dx = \sum_{k \in \mathbb{Z} \cap [a,b]} \varphi(k)$$ $$=\lim_{N \to \infty} \langle 1+2 \sum_{n=1}^N \cos(2\pi n x), \varphi \rangle = \lim_{N \to \infty} \int_{-\infty}^\infty (1+2 \sum_{n=1}^N \cos(2\pi n x)) \varphi(x) \, dx$$
So what I mean is : there is no Fourier series for the Dirac delta, there is only a Fourier series for the Dirac comb $\text{I}\Pi(x)$.
And read a course on : the Fourier series and the Fourier transform, on the distributions, on some complex analysis and the Laplace/Mellin transform.
(*) Since $\text{I}\Pi(x)$ is tempered distribution of order $1$, you can extend $\langle \text{I}\Pi,\varphi \rangle$ to any $\varphi(x)$ continuous (at $x \in\mathbb{Z}$) and with compact support, or decreasing fast enough at $x \to \infty$.
For example, it is perfectly true that $\langle \text{I}\Pi(x) ,x^{-s}\Lambda(\lfloor x+1/2 \rfloor) \rangle = \sum_{n=1}^\infty n^{-s}\Lambda(n) = \frac{-\zeta'(s)}{\zeta(s)}$ for $Re(s) > 1$, but it doesn't mean that $$\frac{-\zeta'(s)}{\zeta(s)} = \lim_{N\to \infty} \int_{1/2}^\infty x^{-s} \Lambda(\lfloor x+1/2 \rfloor)(1+2\sum_{n=1}^N \cos(2\pi n x)) \, dx$$ and this is all the point of the theory of distributions (and the theory of the Riemann zeta function) to study those kind of things.