On the number of representations of an odd integer as a sum of three primes, one of which belongs to an arithmetic progression. (English) Zbl 0911.11048

Tolev, D. I.

Lupanov, O. B. (ed.), Analytic number theory and applications. Collected papers in honor of the sixtieth birthday of Professor Anatolii Alexeevich Karatsuba. Moscow: MAIK Nauka/Interperiodica Publishing, Proc. Steklov Inst. Math. 218, 414-432 (1997) and Tr. Mat. Inst. Steklova 218, 415-432 (1997).

Let \[ J_{k,l}(N)= \sum_{\substack{ p_1+p_2+p_3=N\\ p_1\equiv l\pmod k}} \log p_1\log p_2\log p_3, \] the \(p_i\) running over primes. Then it is shown that \[ \sum_{k\leq N^\theta} \max_{(k,l)=1} \Biggl| J_{k,l}(N)- \frac{N^2}{2\varphi(k)} {\mathfrak S}_{k,l} (N) \Biggr|\ll_A N^2(\log N)^{-A}, \] for any \(A\) and any \(\theta< 1/3\). Here \({\mathfrak S}_{k,l}(N)\) is an appropriately defined singular product. This analogue of the Bombieri-Vinogradov theorem can be combined with sieve methods. The result is motivated by the reviewer’s work [J. Lond. Math. Soc. (2) 23, 396-414 (1981; Zbl 0494.10028)] on three primes and an almost-prime in arithmetic progression.
The author has subsequently improved the range for \(\theta\) to \(\theta<1/2\), in unpublished work.
For the entire collection see [Zbl 0907.00013].

Reviewer: D.R.Heath-Brown (Oxford)

MSC:

11P32	Goldbach-type theorems; other additive questions involving primes
11N13	Primes in congruence classes
11P55	Applications of the Hardy-Littlewood method
11N35	Sieves

Keywords:

sum of three primes; ternary Goldbach conjecture; primes and almost-prime in arithmetic progression; Bombieri-Vinogradov theorem

Citations:

Zbl 0494.10028