On a theorem of Bredihin and Linnik

We give a new proof of a theorem of B.M. Bredihin which was originally proved by extending Linnik's solution, via his dispersion method, of a problem of Hardy and Littlewood.


Introduction
Among the many beautiful consequences of Linnik's dispersion method is an asymptotic formula for the number of solutions to the equation p = a 2 + b 2 + 1 in primes p x and integers a and b. This result of 1965, due to Bredihin [B] was a follow-up to Linnik's celebrated work on the Hardy-Littlewood problem, cf. Chapter 7 of [L]. The involved arguments are lengthy and complicated, though very inventive. Due to much progress over the intervening years, much shorter arguments can now be put forward. This of course does not mean that they are shorter ab-initio. Our purpose here is to illustrate how these arguments can be applied. THEOREM 1. Let S(x) be the number of solutions to (1.1) p = a 2 + b 2 + 1 in integers a and b and primes p ≡ 3(mod 8), p x. We have where the constant c is given by with χ being the Dirichlet character of conductor 4. * Supported in part by NSERC grant A5123. * * Supported in part by NSF grant DMS-1406981. The other reduced residue classes modulo 8 can be covered by essentially the same arguments but we do not treat them. Note that the theorem shows that the integers p − 1 tend to have about as many representations as the sum of two squares as does a typical integer n. Recall also that, if the number of representable p − 1 is counted without multiplicity in a and b, then the order of magnitude is given by x/(log x) 3/2 by a theorem of the second-named author [I].

Dirichlet divisor switching
This is similiar in many respects to the divisor function τ (n). The number of representations of n as the sum of two squares is equal to 4λ(n). If n ≡ 1(mod 4) then, in (2.1), χ(a) can be replaced by χ(b); therefore we can write for any y > 0. We can refine this partition by integrating over y against a smooth weight function. Let w(t) be a smooth function supported on 1 t 2 such that Let Y 1, multiply (2.2) by w(y/Y ) and integrate with the measure y −1 dy, getting Note that if X < n 2X we can choose Y = √ X so the integration in (2.4) runs over the segment 1 2 √ X < y < 2 √ X.

Primes in arithmetic progressions
The key input which greatly streamlines the proof is the main result of [BFI] which gives asymptotics of Bombieri-Vinogradov type for the distribution of primes in arithmetic progressions and which treats moduli of the progression which go beyond the range of that which can be sucessfully handled even on the assumption of the Generalized Riemann Hypothesis.
We state this restricted to a range somewhat lesser than that in [BFI], which is however sufficient for our needs and is conveniently recorded as Theorem 2.2.1 of [FI].
depending only on a and A. We actually require a slightly modified form of (3.1) which follows from it in two easy steps. In the first place we have implied constant depending only on a,k and A. To this end one merely splits the indexed variables into classes modulo k, which is harmless for k fixed.
In the second step we modify (3.2) to a counting of primes with smooth weight.
LEMMA 3.1. Let f (t) be a smooth function supported on 1 t 2. We have the implied constant depending only on a,k, A and f .
Proof. We write Given 1 t 2 this implies p tX. Applying (3.2) with x = tX and integrating the result over t, we derive (3.3).

Proof of the theorem
We have We are going to evaluate for every X 3. Applying (2.4) we write Here we can replace w((p − 1)/2yY ) by w(p/2yY ) up to an error term O(1/yY ) which contributes to T (X) a bounded amount: Note that the integration runs over the segment 1 4 √ X < y < 2 √ X. Now we can apply (3.2) for the first term and (3.3) for the second term with q = b, k = 8, ℓ = 3, getting Next, we replace the sum over b < y by the complete series Therefore, T (X) = 2c 1 π(2X) − π(X) + O X log log X log X 2 .