Cesaro Averages For Goldbach Representations With Summands In Arithmetic Progressions

Let Lambda(n) be the von Mangoldt function, let n >= 2 be an integer and letR-G(n; q,a,b) := Sigma(m1+m2=nm1 equivalent to amodqm2 equivalent to bmodq) Lambda(m(1))Lambda(m(2))be the counting function for the Goldbach numbers with summands in arithmetic progression modulo a common integer q. We prove an asymptotic formula for the weighted average, with Cesaro weight of order k > 1, with k is an element of R, of this function. Our result is uniform in a suitable range for q.