Counting distinct zeros of the Riemann zeta--function

Abstract. Bounds on the number of simple zeros of the derivatives of a function are used to give bounds on the number of distinct zeros of the function. The Riemann ξ-function is defined by ξ(s)=H(s)ζ(s), where H(s) = 1 2s(s−1)π − 1 2 sΓ ( 1 2s)and ζ(s)istheRiemannζ-function. The zeros of ξ(s) and its derivatives are all located in the critical strip 0 <σ<1, where s = σ + it. SinceH(s) is regular and nonzero for σ>0, the nontrivial zeros of ζ(s) exactly correspond to those of ξ(s). Let ρ (j) = β + iγ denote a zero of the j th derivative ξ (j) (s), and denote its multiplicity by m(γ). Define the following counting functions: N (j) (T) = � ρ (j) =β+iγ 1 zeros of ξ (j) (σ + it)with0<t<T N(T)=N (0) (T) zeros of ξ(σ + it)with0<t<T N (j) s (T) = � 1 simple zeros of ξ (j) (σ + it)with0<t<T ρ (j) =β+iγ m(γ)=