Almost primes in almost all very short intervals
Kaisa Matom\"aki

TL;DR
The paper proves that for large enough intervals, almost all contain a number with at most two prime factors, using advanced sieve methods and results on Kloosterman sums.
Contribution
It introduces a new result showing almost all very short intervals contain almost primes, employing Richert's weighted sieve and deep exponential sum estimates.
Findings
Almost all intervals of the form (x - h log X, x] contain an almost prime for large h and X.
The proof combines sieve techniques with bounds on Kloosterman sums.
The result extends understanding of the distribution of almost primes in short intervals.
Abstract
We show that as soon as with , almost all intervals with contain a product of at most two primes. In the proof we use Richert's weighted sieve, with the arithmetic information eventually coming from results of Deshouillers and Iwaniec on averages of Kloosterman sums.
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Taxonomy
TopicsAnalytic Number Theory Research
