## 8 Citations

Short intervals containing a prescribed number of primes

- Mathematics, Computer ScienceJournal of Number Theory
- 2019

A Note on the Distribution of Primes in Intervals

- Mathematics
- 2018

Assuming a certain form of the Hardy–Littlewood prime tuples conjecture, we show that, given any positive numbers λ1, …, λr and nonnegative integers m1, …, mr, the proportion of positive integers \(n…

Almost primes in almost all short intervals

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2016

Abstract Let Ek be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x + log1+ϵ x] contain E 3 numbers, and almost all intervals [x,x + log3.51 x]…

Almost Primes in Almost All Short Intervals

- Mathematics
- 2015

Let Ek be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x+ log x] contain E3 numbers, and almost all intervals [x, x + log x] contain E2 numbers.…

POSITIVE PROPORTION OF SHORT INTERVALS CONTAINING A PRESCRIBED NUMBER OF PRIMES

- MathematicsBulletin of the Australian Mathematical Society
- 2019

We prove that for every $m\geq 0$ there exists an $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}(m)>0$ such that if $0<\unicode[STIX]{x1D706}<\unicode[STIX]{x1D700}$ and $x$ is sufficiently large in…

Weighted Average Number of Prime $m$-tuples lying on an Admissible $k$-tuple of Linear Forms

- Mathematics, Computer Science
- 2018

An upper bound is found for the sum of $\sum_{x<n\leq 2x}\textbf{1}_{\mathbb{P}}(n+h_{i_{1}})w_{n}), which will depend on an integral of a smooth function and on the singular series of $\mathcal{H}$, which naturally arises in this context.

The twin prime conjecture

- MathematicsJapanese Journal of Mathematics
- 2019

The Twin Prime Conjecture asserts that there should be infinitely many pairs of primes which differ by 2. Unfortunately this long-standing conjecture remains open, but recently there has been several…

GAPS BETWEEN PRIMES

- MathematicsProceedings of the International Congress of Mathematicians (ICM 2018)
- 2019

We discuss recent advances on weak forms of the Prime $k$-tuple Conjecture, and its role in proving new estimates for the existence of small gaps between primes and the existence of large gaps…

## References

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Dense clusters of primes in subsets

- Mathematics, Computer ScienceCompositio Mathematica
- 2016

It is proved that any subset of the primes which is ‘well distributed’ in arithmetic progressions contains many primesWhich are close together, and bounds hold with some uniformity in the parameters.

Primes in tuples I

- Mathematics
- 2009

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the…

ON THE DISTRIBUTION OF PRIMES IN SHORT INTERVALS

- Mathematics
- 2010

One of the formulations of the prime number theorem is the statement that the number of primes in an interval (n, n + ft], averaged over n ^ JV, tends to the limit A, when JV and h tend to infinity…

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…

Topics in Multiplicative Number Theory

- Mathematics
- 1971

Three basic principles.- The large sieve.- Arithmetic formulations of the large sieve.- A weighted sieve and its application.- A lower bound of Roth.- Classical mean value theorems.- New mean value…

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…

Goldbach versus de Polignac numbers

- Mathematics
- 2015

In this article we prove a second moment estimate for the Maynard-Tao sieve and give an application to Goldbach and de Polignac numbers. We show that at least one of two nice properties holds. Either…

Über die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind.

- Commentat. Phys.-Math. 5(25):1–37,
- 1931