Questions tagged [prime-gaps]

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What is the best unconditional upper bound for $\vert\log\frac{\pi_{2k}(x)}{\pi_{2l}(x)}\vert$?

Let for a positive integer $m$ denote by $\pi_{2m}(x)$ the quantity $\sharp\{n\leq x, p_{n+1}-p_{n}=2m\}$. What is the best unconditional upper bound in terms of $k$, $l$ and $x$ one can get for $\...
1
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0answers
100 views

Primes in this region

Let $q \geq 5$ be a prime number, and consider : $N_q = \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize p}}$ Using Chinese remainder theorem we can show that : $$\#\{(...
5
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0answers
223 views

Symmetry of the distribution of prime gaps

Following Positive proportion of logarithmic gaps between consecutive primes let for given $\lambda$, $\alpha$ and for any $x$ all positive the quantities $S^{-}_{\lambda,\alpha}(x):=\#\{p_{n+1}\...
5
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3answers
637 views

Positive proportion of logarithmic gaps between consecutive primes

For $x, \lambda > 0$, define $$S_\lambda(x) := \#\{p_{n+1} \leq x : p_{n+1} - p_n \geq \lambda \log x\} ,$$ where $p_n$ is the $n$th prime number. It is known [1] that an uniform version of the ...
6
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0answers
99 views

Upper bound for number of primes close to the next prime

Let $p_n$ denote the $n$th prime number and let $g_n := p_{n+1} - p_n$ be the $n$th prime gap. I'm looking for a good upper bound for the quantity $$G(x, y) :=\#\{p_n \leq x : g_n \leq y\} ,$$ holding ...
26
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0answers
549 views

Does this infinite primes snake-product converge?

This re-asks a question I posed on MSE: Q. Does this infinite product converge? $$ \frac{2}{3}\cdot\frac{7}{5}\cdot\frac{11}{13}\cdot\frac{19}{17}\cdot\frac{23}{29}\cdot\frac{37}{31} \cdot \cdots \...
5
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1answer
337 views

consecutive prime gaps and explicit bound

I am aware of the theorem that $p_{n+1}- p_n \leq n^{0.525}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" ...
6
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2answers
839 views

$\pi((n+1)^2)-\pi(n^2) \le \pi(n)$ for all $n \ge 370$?

There are some conjectures of the form: There always exist at least $X$ prime numbers between $A$ and $B$. Examples: Bertrand's postulate: for every $n>1$ there is always at least one prime $p$ ...
3
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0answers
116 views

Number of prime differences

Has any progress been made since Chen on bounding \begin{equation*} G(n) = \#\{\epsilon N < p_1, p_2 \leq N: n = p_1 - p_2\} \end{equation*} from above? As far as I can tell, the best upper ...
4
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1answer
416 views

Moments of merit

The merit of a prime gap equals $(p_{n+1}-p_n)/\ln p_n$. One can interrogate the statistics of merit by first restricting $n<M$ for some $M$, and then letting $M$ approach $\infty$. The very ...