Geodesic
Generalization of Goldbach's ternary problem with almost equal terms
Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 264-298

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An asymptotic formula is obtained for the number of representations of a sufficiently large natural $N$ in the form $b_1p_1+b_2p_2+b_3p_3=N$ with the conditions $$ \left|b_ip_i-\frac{N}3\right|\le H, H\ge (b_1b_2b_3)^\frac43N^\frac23(\ln N)^{60}, b_i\le(\ln N)^{B_i}, $$ where $b_1$, $b_2$ $b_3$, $N$ are pairwise coprime natural numbers, $B_i$ — arbitrary fixed positive numbers.
Keywords: ternary Goldbach problem, almost equal terms, short exponential sum with primes, small neighborhood of centers of major arcs. 
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     author = {Z. Kh. Rakhmonov and I. Allakov and B. Kh. Abrayev},
     title = {Generalization of {Goldbach's} ternary problem with almost equal terms},
     journal = {\v{C}eby\v{s}evskij sbornik},
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Z. Kh. Rakhmonov; I. Allakov; B. Kh. Abrayev. Generalization of Goldbach's ternary problem with almost equal terms. Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 264-298. http://geodesic.mathdoc.fr/item/CHEB_2023_24_4_a15/