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Results tagged with prime-numbers
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user 17218
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
6
votes
1
answer
171
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Is the set of all solutions $x > 0$ to $ \pi(x) = \operatorname{li}(x)$ unbounded?
Is the set of all solutions $x > 0$ to the equation $\pi(x) = \operatorname{li}(x)$ unbounded? Is $\liminf_{x \to \infty} |\pi(x)-\operatorname{li}(x)|$ equal to $0$?
Here, $\pi(x)$ denotes the prime …
- 4,985
4
votes
1
answer
387
views
Bounding integrals involving $\operatorname{li}(x)-\pi(x)$
Let $x >0$. How can one find good $O$ bounds on the integrals
$$\int_0^x\frac{\operatorname{li}(t)-\pi(t)}{t}dt$$
and
$$\int_x^\infty\frac{\operatorname{li}(t)-\pi(t)}{t^2}dt$$
where $\pi(x)$ is the …
- 4,985
3
votes
0
answers
409
views
Proof of an explicit formula for $\pi_0(x)$
Let $\pi(x)$ denote the prime counting function and $$\pi_0(x) = \lim_{\epsilon \to 0} \frac{\pi(x+\epsilon)+\pi(x-\epsilon)}{2}.$$
I've seen noted in a few references the explicit formula
$$\pi_0(x) …
- 4,985
2
votes
0
answers
342
views
An approximation for the prime counting function
NOTE: I've edited the question one last time, to be much simpler, in the hopes of getting more responses.
SETUP: Let $p_n$ denote the $n$th prime, let $p_x = p_{\lceil x \rceil}$ for all $x > 0$, let …
- 4,985
4
votes
0
answers
159
views
On the asymptotic $\pi(x+h(x)) - \pi(x) \sim \frac{h(x)}{\log x} \ (x \to \infty)$
Let $h(x)$ be a function that is positive on $\mathbb{R}_{>0}$ and satisfies $h(x) = o(x)$ and $(\log x)^a = o(h(x))$ for all $a > 0$, as $x \to \infty$. Is it reasonable to expect under these condi …
- 4,985
6
votes
Accepted
Primes which are safe and Sophie Germain
This is asking for the density of Cunningham chains of the first kind of length three. Take the integer polynomials $f_1(n) = n$, $f_2(n) = 2n+1$ and $f_3(n) = f_2(f_2(n)) = 4n+3$ and apply the (rema …
- 4,985