Geodesic
On the Partition of an Odd Number into Three Primes in a Prescribed Proportion
Matematičeskie zametki, Tome 106 (2019) no. 1, pp. 95-107

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We prove that, for any partition $1=a+b+c$ of unity into three positive summands, each odd number $n$ can be subdivided into three primes $n=p_a(n)+p_b(n)+p_c(n)$ so that the fraction of the first summand will approach $a$, that of the second, $b$, and that of the third, $c$ as $n \to \infty$.
Keywords: Goldbach–Vinogradov theorem, distribution of primes, Hardy–Littlewood circle method, trigonometric sums. 
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     author = {A. A. Sagdeev},
     title = {On the {Partition} of an {Odd} {Number} into {Three} {Primes} in a {Prescribed} {Proportion}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {95--107},
     publisher = {mathdoc},
     volume = {106},
     number = {1},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2019_106_1_a8/}
}
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A. A. Sagdeev. On the Partition of an Odd Number into Three Primes in a Prescribed Proportion. Matematičeskie zametki, Tome 106 (2019) no. 1, pp. 95-107. http://geodesic.mathdoc.fr/item/MZM_2019_106_1_a8/