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 $k$. 
@article{MZM_1996_59_4_a9,
     author = {N. M. Timofeev and M. B. Khripunova},
     title = {The {Titchmarsh} problem with integers having a~given number of prime divisors},
     journal = {Matemati\v{c}eskie zametki},
     pages = {586--603},
     publisher = {mathdoc},
     volume = {59},
     number = {4},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_59_4_a9/}
}
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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/