Linked Questions

15
votes
4answers
670 views

$\# \{\text{primes}\ 4n+3 \le x\}$ in terms of $\text{Li}(x)$ and roots of Dirichlet $L$-functions

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 ...
8
votes
2answers
361 views

Tying some pieces regarding the Zeta Function and the Prime Number Theorem together

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

What is the distribution of primes modulo $n$?

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 ...
6
votes
1answer
554 views

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 ...
4
votes
1answer
124 views

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 ...
4
votes
1answer
148 views

Squeezing $\pi(x)$ out of $\psi(x)$

Can $\pi(x)$ be written in terms of $\psi(x)$? I can only seem to approximate it: $$ ...
3
votes
1answer
310 views

Is $\frac1\pi \arctan \frac\pi{\ln x}- \frac1{\ln x}$ related to the trivial solutions $\zeta(-2n)$?

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 $ ...
6
votes
1answer
117 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: ...
3
votes
0answers
370 views

An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

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} ...
7
votes
1answer
166 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) = ...
3
votes
2answers
106 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 ...
2
votes
2answers
91 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 ...
1
vote
0answers
115 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 ...
0
votes
0answers
93 views

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 ...