# Limit points and long gaps between primes

@article{Baker2015LimitPA, title={Limit points and long gaps between primes}, author={Roger C. Baker and Tristan Freiberg}, journal={arXiv: Number Theory}, year={2015} }

Let $d_n = p_{n+1} - p_n$, where $p_n$ denotes the $n$th smallest prime, and let $R(T) = \log T \log_2 T\log_4 T/(\log_3 T)^2$ (the "Erd{\H o}s--Rankin" function). We consider the sequence $(d_n/R(p_n))$ of normalized prime gaps, and show that its limit point set contains at least $25\%$ of nonnegative real numbers. We also show that the same result holds if $R(T)$ is replaced by any "reasonable" function that tends to infinity more slowly than $R(T)\log_3 T$. We also consider "chains" of…

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## References

SHOWING 1-10 OF 13 REFERENCES

Long gaps between primes

- Mathematics, Computer Science
- 2014

The main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the R\"odl nibble method.

Large gaps between consecutive prime numbers

- Mathematics
- 2014

Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log…

Large gaps between primes

- Mathematics
- 2014

We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our…

On the distribution of gaps between consecutive primes

- Mathematics
- 2014

Erd\"os conjectured that the set J of limit points of d_n/logn contains all nonnegative numbers, where d_n denotes the nth primegap. The author proved a year ago (arXiv: 1305.6289) that J contains an…

A note on the distribution of normalized prime gaps

- Mathematics
- 2015

Let us denote the nth difference between consecutive primes by d_n. The Prime Number Theorem clearly implies that d_n is logn on average. Paul Erd\H{o}s conjectured about 60 years ago that the…

Gaps between prime numbers

- Mathematics
- 1988

Let dn = Pn+1 -Pn denote the nth gap in the sequence of primes. We show that for every fixed integer k and sufficiently large T the set of limit points of the sequence {(dn/logn,. .. Idn+k1l/logn)}…

Small gaps between primes

- Mathematics
- 2013

We introduce a renement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This renement avoids previous limitations of the method and allows us to show that for each…

The Distribution of Prime Numbers

- MathematicsNature
- 1933

THIS interesting “Cambridge Tract” is concerned mainly with the behaviour, for large values of x, of the function n(x), which denotes the number of primes not exceeding x. The first chapter gives…

On limit points of the sequence of normalized prime gaps

- Mathematics
- 2016

Let pn denote the n th smallest prime number, and let L denote the set of limit points of the sequence {(pn+1−pn)/logpn}n=1∞ of normalized differences between consecutive primes. We show that, for…

Multiplicative Number Theory

- Mathematics
- 1967

From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The…