Analogues of the Gauss-Vinogradov formula on the critical line

An asymptotic behavior of the sum[Figure not available: see fulltext.] for X → ∞ is studied in the critical strip, where L(s, X_p) is the Dirichlet series with the quadratic character X_p modulo p, where p is a prime number; v=1 or 3. With the help of large seive estimates a formula for this sum is obtained with two asymptotic terms on the critical line of the variable s. As a corollary the asymptotic expansion of this sum at the point s=1/2 is presented. The asymptotic formula for the sum[Figure not available: see fulltext.], where d runs over discriminants of quadratic fields, is also obtained.