On some Dirichlet series related to the Riemann zeta function, I

On the assumption of the truth of the Riemann hypothesis for the Riemann ze ta function we construct a class of modified von-Mangoldt functions with sl ightly better mean value properties than the well known function Lambda. Fo r every epsilon epsilon (0, 1/2) there is a Lambda : N --> C such that i) Lambda(n) = Lambda(n)(1 + O(n(-1/4) log n)) and ii) Sigma(n less than or equal to x) Lambda(n) (1-(n)/(x) =) (x)/(2) + O(x( 1/4+epsilon))(x greater than or equal to 2). Unfortunately, this does not lead to an improved error term estimation for the unweighted sum Sigma(n less than or equal to x) Lambda(n), which would be of importance for the distance between consecutive primes.