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Results tagged with prime-numbers
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user 37555
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.
10
votes
Is Li(x) the best possible approximation to the prime-counting function?
Whether for a finite set $\mathcal{R}$ of roots the approximation
$$
\pi(x)\approx\mathrm{Li}(x)-\frac{1}{2}\mathrm{Li}(x^{1/2})-\sum_{\rho\in\mathcal{R}}\mathrm{Li}(x^\rho)
$$
is "on average" better …
- 5,089
3
votes
Best provable and unconditional lower and upper bounds for Brun's constant
The best known bounds seem to be due to Nicely [``A new error analysis for Brun's constant,'' Virginia J. Sci. 52 (2001), no. 1, 45–55), who showed that Brun's constant is
$$
1.9021605823 \pm 0.000000 …
- 18.4k
22
votes
Accepted
What keeps asymptotic Goldbach's conjecture out of reach of current technology?
As far as I know there are two approaches to Goldbach type problems, the circle method and sieve methods. In the sequel I will restrict myself to the circle method, hoping that someone else writes som …
- 5,407
10
votes
Can anything deep be said uniformly about conjectures like Goldbach's?
There exist surprising counterexamples. Elsholtz and Dietmann found the following: If $p\equiv 7\pmod{8}$ is prime, then the equation $x^2+y^2+z^4=p^2$ has no non-trivial solution. You might argue tha …
2
votes
Accepted
Density of integers with a large rough divisor
If $a/b$ is not too large, you can compute the probability using arguments as in the computation leading to the asymptotics for smooth numbers. In theory, you can compute for all $\beta, \gamma$ a rea …
3
votes
Goldbach's conjecture for the Liouville function
In "The equation $\omega(n)=\omega(n+1)$" (Mathematika 50, 99-101, 2003) it was shown that the equation $\omega(n)=\omega(n+1)$ has infinitely many solutions. Pintz has a series of results on consecut …
1
vote
Spacing of fractions with prime denominator
Wolke (On the large sieve with primes, Acta Math. Acad. Sci. Hungar. 22 (1971/72), 239–247, MathSciNet MR0291121 (45 #215)) has worked on this question. His motivation was Gallagher's approach to the …
10
votes
Does the Prime Number Theorem have anything to do with Erdos-Kac law or vice versa?
The right context of your question is probably the area of Beurling primes. An arithmetic semigroup is a semigroup $(G,\cdot)$ together with a norm $\|\cdot\|:G\rightarrow[1,\infty)$, such that $\|gh\ …
- 4,713
1
vote
Writing integers as determinants of matrices with prime entries.
Goldston, Graham Pintz and Yildirim ( https://arxiv.org/abs/0803.2636 ) showed that among three linear forms $\ell_1, \ell_2, \ell_3$, which satisfy the obvious local conditions, there are two, say $\ …
5
votes
Accepted
Specializing non-trivial primality tests
Checking whether an integer $n$ has a divisor in a given interval is essentially equivalent to factorization. Factoring is essentially equivalent to finding the smallest prime factor. So suppose that …
12
votes
0
answers
626
views
Sieve bound for prime $k$-tuples
Let $d_1<d_2<\dots<d_k$ be integers. Then the number of integers $n\leq x$, such that $n+d_1, n+d_2, \ldots, n+d_k$ are simultaneously prime, is bounded above by
$$
\mathfrak{S}(d_1, \ldots, d_k) (Ck) …
- 105k
14
votes
Understanding Vaughan's Identity
The analytic version of Vaughan's identity is
$$
\frac{\zeta'}{\zeta} = F+\zeta'G-FG\zeta + \left(\frac{\zeta'}{\zeta}-F\right)(1-\zeta G).
$$
Here the last factor to the right is the most complicated …
4
votes
Shifted primes avoiding a set of divisors
No, $\delta_y$ need not tend to 0. Take a rapidly increasing sequence of integers $y_n$. Then define the set $B$ as $\{p-1|\exists n: y_n\leq p\leq 2y_n\}$. Then we have
\begin{eqnarray*}
\delta_{y_n} …
8
votes
Primality test for $2p+1$
The if-part is a special case of Pocklingtons theorem (see https://en.wikipedia.org/wiki/Pocklington_primality_test ). The only if part requires some computation, as you need a quadratic non-reside mo …
4
votes
Accepted
Numbers related to the Riemann hypothesis
Proving such results falls into three parts. First you take an effective version of the prime number theorem, which implies all your desired bound for sufficient. Second you write a computer program ( …