On the Zeros of Higher Derivatives of HardyʹsZ-Function Original Research Article

Letkbe any positive integer andN0, k(T) the number of the zeros in the interval (0, T) ofZ(k)(t), thekth derivative of HardyʹsZ-function. We prove an inequality forN0, k(T) (Theorem 1), and also prove that it can be replaced by the equality under the Riemann hypothesis (Theorem 2). The key fact of the proof is the construction of a meromorphic functionηk(s), which satisfies an appropriate recurrence formula and a functional equation.