Nonlinear sums of characters over primes (Q542209)

Suppose that \(q\) is an odd prime, \(\chi_1(n)\) and \(\chi_2(n)\) are nonprincipal Dirichlet characters modulo \(q\), \(a\) and \(b\) are integers, \(a\not\equiv b \pmod q\), \(q^{0.75+\varepsilon}\leq X\leq q^2\). The author proves that \[ |\sum_{p\leq X}\chi_1(p+a)\chi_2(p+b)|\leq c_1(\varepsilon)Xq^{-0.05\varepsilon^2}. \] Let \(K(X)\) denote the number of primes \(p\) not exceeding \(X\) for which \(p+a\) and \(p+b\) are primitive roots modulo \(q\). If \(ab\not\equiv 0(\bmod q)\) then \[ K(X)=\left(\frac{\varphi(q-1)}{q-1}\right)^2\pi(X)+O(Xq^{-0.04\varepsilon^2}), \] where the constant in \(O\) depends only on \(\varepsilon\). Suppose that \(Q\) is a positive integer (common difference of a progression), \(1\leq Q\leq q^A,0\leq l<Q,(l,Q)=1\), and \(A\) and \(B\) are arbitrary constants such that \(A\geq 1\) and \(B\geq 1\). If \[ Q^3q^{0.75+\varepsilon}\leq X\leq Q^3q^B \] then \[ |\sum_{{p\leq X\atop p\equiv l\pmod Q}}\chi_1(p+a)\chi_2(p+b)|\leq c_2(\varepsilon)Q^{-1}Xq^{-0.005\varepsilon^2}. \]