<Dl> <Dd> ∑ n = M + 1 M + N (n q) <4 π 2 q log ⁡ q + 0.41 q + 0.61 . (\ displaystyle \ left \ sum _ (n = M + 1) ^ (M + N) \ left ((\ frac (n) (q)) \ right) \ right <(\ frac (4) (\ pi ^ (2))) (\ sqrt (q)) \ log q + 0.41 (\ sqrt (q)) + 0.61 .) </Dd> </Dl> <Dd> ∑ n = M + 1 M + N (n q) <4 π 2 q log ⁡ q + 0.41 q + 0.61 . (\ displaystyle \ left \ sum _ (n = M + 1) ^ (M + N) \ left ((\ frac (n) (q)) \ right) \ right <(\ frac (4) (\ pi ^ (2))) (\ sqrt (q)) \ log q + 0.41 (\ sqrt (q)) + 0.61 .) </Dd> <P> Montgomery and Vaughan improved this in 1977, showing that, if the generalized Riemann hypothesis is true then </P> <Dl> <Dd> ∑ n = M + 1 M + N χ (n) = O (q log ⁡ log ⁡ q). (\ displaystyle \ left \ sum _ (n = M + 1) ^ (M + N) \ chi (n) \ right = O \ left ((\ sqrt (q)) \ log \ log q \ right).) </Dd> </Dl>

The integer y is a quadratic residue modulus n and its square roots are known