Linked Questions

In a paper about Prime Number Races, I found the following (page 14 and 19): This formula, while widely believed to be correct, has not yet been proved. $$ \frac{\int\limits_2^x{\frac{dt}{\ln t}...

According to Riemann (I think) the (exact) prime counting function is given by: $$ \pi(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = \...

I came across two remarks that I would appreciate help in making the connections. I) In Riemann's Explicit Formula: for $x > 1$ $\Pi = Li(x) - \sum_{\rho:\zeta(\rho)=0}Li (x^{\rho})- \log(2) +$ ...

Let $n\geq 2$ and let $k$ be "considerably larger" than $n$ (like some large multiple of $n$). Then for each $i$ such that $0<i<n$ and $\gcd(i,n)=1$ let's define $$c_i=\left|\{p_j\;|\; p_j\equiv ...

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/...

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 ...

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(...

I think, here, I found $$ P_\color{red}x(\color{blue}s)=\sum_{p<\color{red}x} \frac{1}{p^{\color{blue}s}} =\sum_{\color{green}n=1}^{\infty}\frac{ \mu (\color{green}n)}{\color{green}n} \sum_{z\in\{...

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 $-\frac{\zeta'(s)}{\zeta(...

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: ...

Can $\pi(x)$ be written in terms of $\psi(x)$? I can only seem to approximate it: $$ \pi(x)\approx\sum_{n=1}^{\infty}\left[\dfrac{\mu(n)}{n}\left(\dfrac{1}{\log(x^{1/n})}\left(\psi(x^{1/n})-x^{1/n}+\...

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 $ \...

I was reading some articles related to Euler sums and the Riemann zeta function, when I came across this definition: $$ L(n,\chi_4) = \sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^n} $$ What is this ...

The explicit formula expresses the deep connection between the primes $p$ and the non-trivial zeros $\rho$ of $\zeta(s)$. The prime-counting function is given by the following formula giving primes in ...

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 several ...