Primes in short arithmetic progressions with rapidly increasing differences (Q2719044)

The author establishes the following mean value theorem of Bombieri-Vinogradov type. Let \(f\) be a polynomial of degree \(n\geq 1\), with integer coefficients, and positive leading coefficient. If \(4\beta n<1\) and \(A>0\), then NEWLINE\[NEWLINE\sum_{D\leq x^\beta} \frac{\varphi(f(D))}{D} \max_{(r,f(D))=1} \max_{y\leq x} \left|\psi(y,f(D),r)- \frac{y}{\varphi(f(D))} \right|\ll x(\log x)^{-A}.NEWLINE\]NEWLINE This is remarkable in that the moduli \(f(D)\) run through a very thin sequence. In the linear case it is of course weaker than the well-known Bombieri-Vinogradov theorem. The proof follows in principle Linnik's dispersion method. The problem is transferred to the estimation of bilinear forms by a Vaughan-Heath-Brown identity.