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### Summing over General Functions of Primes and an Application to Prime $\zeta$ Function

Along the lines of thought given here, is it in general possible to substitute a summation over a function $f$ of primes like the following: $$\sum_{p\le x}f(p)=\int_2^x f(t) d(\pi(t))\tag{1}$$ and ...
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### How does it follow $s\int_1^{\infty}\frac{\psi(x)}{x^{s+1}}dx$?

I have two relations: 1)$-\frac{\zeta'(s)}{\zeta(s)}=\sum_{1}^{\infty}\frac{\Lambda(n)}{n^s}$. 2)$\psi(x)=\sum_{n\leq x}\Lambda(n)$. From these two how does it follow that ...
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### State of art of prime numbers distribution [closed]

I was reading some questions about prime numbers posted in latest days and a question came to my mind: What is the state of art of the research into prime numbers distribution? I read then ...
Dirichlet's theorem on arithmetic progressions states that for any two positive integers a and b, if gcd(a,b) = 1 then the arithmetic progression $t(x)=ax+b$ $(x ≥ 0)$ contains infinitely many prime ...
### Will this work as a $\pi_{4n\pm 1}$ prime counting function?
I forgot where it was, but I remember someone saying that you need $\phi(4)$, which is two, of Dirichlet's $L$ functions to get a prime counting function for primes of the form $4n\pm 1$ less than ...