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

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1/2$$, in unpublished work.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].

MSC:

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

Zbl 0494.10028