Almost primes in almost all short intervals
2016 ◽
Vol 161
(2)
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pp. 247-281
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AbstractLet Ek be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x + log1+ϵx] contain E3 numbers, and almost all intervals [x,x + log3.51x] contain E2 numbers. By this we mean that there are only o(X) integers 1 ⩽ x ⩽ X for which the mentioned intervals do not contain such numbers. The result for E3 numbers is optimal up to the ϵ in the exponent. The theorem on E2 numbers improves a result of Harman, which had the exponent 7 + ϵ in place of 3.51. We also consider general Ek numbers, and find them on intervals whose lengths approach log x as k → ∞.
1998 ◽
Vol 124
(1)
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pp. 1-14
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1997 ◽
Vol 122
(2)
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pp. 193-205
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2000 ◽
Vol 157
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pp. 103-127
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1992 ◽
Vol 44
(6)
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pp. 1121-1154
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1986 ◽
Vol 22
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pp. 289-307
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1989 ◽
Vol 13
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pp. 387-401
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2020 ◽
Vol 2020
(763)
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pp. 1-24
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