# Linked Questions

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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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The Prime Counting Function $\pi(x)$ is given $$\pi(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x} ,$$ with $\... 1answer 146 views ### The use of log in the Mean density of the nontrivial zeros of the Riemann zeta function (part 2) As part of my MSc I am reviewing a paper. The paper is a review on the statistical distribution of the unfolded zeros (see below) of the Reimann functional equation. In the paper there is a sentence: ... 1answer 207 views ### If these two expressions for calculating the prime counting function are equal, why doesn't this work? So I've seen some different explanations of how the zeros of the zeta function can predict the prime counting function. The common example is that $$\pi(x)=\sum_{n=1}^\infty \frac{\mu(n)}{n}J(x^{1/... 1answer 205 views ### Is there a complex variant of Möbius' function? When you're dealing with arithmetic functions, you might have come across the classical Möbius' function$$ \mu(n)=\begin{cases} (-1)^{\omega(n)}=(-1)^{\Omega(n)} &\mbox{if }\; \omega(n) = \Omega(... 2answers 122 views ### 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 ... 2answers 113 views ### Equality involving$\sum_n \sin(\gamma_n \log x)/\gamma_n$This is I think an algebra confusion about an equality of Littlewood, $$\frac{\psi(x) - x}{\sqrt{x}} = -2\sum_{1}^{\infty}\frac{\sin( \gamma_n\log x)}{\gamma_n} + O(1).\hspace{20mm}(1)$$ He refers ... 0answers 207 views ### Approximate zeros of a (hypothetical) analog of$\zeta(s)$[Added numbers 11/13.] Motivation (can skip). When prime powers$p_n$are used to calculate $$y(x) = \sum_{n=1}^{N}\frac{\sin (x \log p_n)}{p_n},\hspace{5mm}(1)$$ for (say)$N= 30,x>5$, at ... 0answers 128 views ### Dirichlet's Theorem 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 ...

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