# 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 $\...

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### 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 :
$$\#\{(...

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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}\...

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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 ...

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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 ...

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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 \...

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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" ...

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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 $p$ ...

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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 ...

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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 ...