Questions tagged [prime-gaps]

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1 answer
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Numerical estimates for a function relating to twin primes :

Consider the following function : $$F(s)= \sum_{\text{$p,\ p+2$ are primes}} \left({\frac{1}{p^s}}+{\frac{1}{(p+2)^s}}\right).$$ Brun's theorem tells us that $F(1)$ is finite. We are looking for ...
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1 vote
2 answers
141 views

Is the abscissa of convergence of $s\mapsto\sum_{n>0}(ng_{n}/2)^{-s}$ known?

The famous Polignac conjecture posits that the $n$-th prime gap $g_{n}:=p_{n+1}-p_{n}$ attains all even positive integral values infinitely many times, which implies $\displaystyle{\zeta_{Pol}:=s\...
Sylvain JULIEN's user avatar
-1 votes
1 answer
242 views

A conjecture about an inequality that involve Ramanujan primes

In this post we denote for integers $n\geq 1$ the $n$-th Ramanujan prime as $R_n$ (thus the sequence A104272 from the On-Line Encyclopedia of Integer Sequences), I add a conjecture that I think can be ...
user142929's user avatar
0 votes
0 answers
104 views

Variants of Nicholson's inequalities for prime numbers, involving the Lambert $W$ function

The purpose of this post is ask about two closely related/inspired conjectures from inequalities due to Nicholson (see [1]) and Visser [2]. If my reasonings are right should be stronger versions of ...
user142929's user avatar
-4 votes
3 answers
654 views

Remarkable articles about the distribution of prime numbers that were written by contemporary physicists [closed]

I would like to ask about if you know papers containing remarkable achievements that were written by contemporary physicists concerning the distribution of prime numbers (or closely related, maybe the ...
3 votes
1 answer
335 views

Is there a clear criterion or rule about when one can use the heuristic given by Cramér random model for prime numbers?

I don't know if I have misunderstood the information from the section about the heuristic related to Cramér's conjecture and the known facts about Maier's theorem, from the Wikipedia Cramér's ...
user142929's user avatar
4 votes
1 answer
500 views

A weaker version of the Brocard's Conjecture

Brocard's conjecture states that: If $p_{k}$ and $p_{k+1}$ are consecutive prime numbers greater than $2$, then between $p_{k}²$ and $p_{k+1}²$ there are at least four prime numbers. I know that is ...
Safwane's user avatar
  • 963
20 votes
2 answers
2k views

Is every prime the largest prime factor in some prime gap?

Definition: In the gap between any two consecutive odd primes we have one or more composite numbers. One of these composite number will have a prime factor which is greater than that of any other ...
Nilotpal Kanti Sinha's user avatar
0 votes
1 answer
218 views

Is this theorem on the abundance of prime patterns/k-tuples known?

I am looking for references regarding the following statement. For any two natural numbers x and y there must be a prime k-tuple (a, b, ...) corresponding to x consecutive primes (n+a, n+b, ...) for ...
Thomas Traill's user avatar
1 vote
1 answer
190 views

A question about a sum that involves gaps between twin primes, on assumption of the First Hardy–Littlewood conjecture

I wondered, inspired in a result mentioned from [1] (page 45), what should be the asymptotic behaviour of the sequence on assumption of the First Hardy–Littlewood conjecture $$\sum_{\substack{\text{...
user142929's user avatar
-3 votes
1 answer
253 views

A Bonse's inequality for semiprimes, with a good mathematical content

A semiprime $s$ is a positive integer that is the product of two prime numbers, see Semiprine the encyclopedia Wikipedia. A well-known inequality, with applications, that involves prime numbers is the ...
user142929's user avatar
1 vote
0 answers
277 views

Prime numbers 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 : $$\#\{(...
Lagrida Yassine's user avatar
5 votes
0 answers
281 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}\...
Sylvain JULIEN's user avatar
5 votes
3 answers
794 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 ...
Kello's user avatar
  • 113
6 votes
0 answers
132 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 ...
Kello's user avatar
  • 113
28 votes
0 answers
695 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 \...
Joseph O'Rourke's user avatar
5 votes
1 answer
423 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" ...
Ahmad's user avatar
  • 595
6 votes
2 answers
1k 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 $p$ ...
Đào Thanh Oai's user avatar
3 votes
0 answers
125 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 ...
Annemie's user avatar
  • 71
4 votes
1 answer
445 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 ...
David Feldman's user avatar

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