Geodesic
The Titchmarsh problem with integers having a~given number of prime divisors
Matematičeskie zametki, Tome 59 (1996) no. 4, pp. 586-603.

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The asymptotics for the number of representations of $N$ as $N\to\infty$ is expressed as the sum of a number having $k$ prime divisors and a product of two natural numbers. The asymptotics is found for $k\le(2-\varepsilon)\ln\ln N$ and $(2+\varepsilon)\ln\ln N\le k\le b\ln\ln N$, where $\varepsilon>0$. The results obtained are uniform with respect to $k$. 
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N. M. Timofeev; M. B. Khripunova. The Titchmarsh problem with integers having a~given number of prime divisors. Matematičeskie zametki, Tome 59 (1996) no. 4, pp. 586-603. http://geodesic.mathdoc.fr/item/MZM_1996_59_4_a9/

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