Counting distinct zeros of the Riemann zeta-function

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) = 2s(s−1)π 1 2 Γ( 1 2 s) and ζ(s) is the Riemann ζ-function. The zeros of ξ(s) and its derivatives are all located in the critical strip 0 < σ < 1, where s = σ+ it. Since H(s) is regular and nonzero for σ > 0, the nontrivial zeros of ζ(s) exactly correspond to those of ξ(s). Let ρ = β + iγ denote a zero of the j derivative ξ(s), and denote its multiplicity by m(γ). Define the following counting functions: N (T ) = ∑ ρ(j)=β+iγ 1 zeros of ξ(σ + it) with 0 < t < T N(T ) = N (T ) zeros of ξ(σ + it) with 0 < t < T N (j) s (T ) = ∑ ρ(j)=β+iγ m(γ)=1 1 simple zeros of ξ(σ + it) with 0 < t < T N (j) s, 2 (T ) = ∑ ρ(j)= 1 2 +iγ m(γ)=1 1 simple zeros of ξ(12 + it) with 0 < t < T Mr(T ) = ∑ ρ(0)=β+iγ m(γ)=r 1 zeros of ξ(σ + it) of multiplicity r with 0 < t < T M≤r(T ) = ∑ ρ(0)=β+iγ m(γ)≤r 1 zeros of ξ(σ + it) of multiplicity ≤ r with 0 < t < T where all sums are over 0 < γ < T , and zeros are counted according to their multiplicity. It is well known that N (T ) ∼ 1 2πT logT . Let αj = lim inf T→∞ N (j) s, 2 (T ) N (j)(T ) . βj = lim inf T→∞ N (j) s (T ) N (j)(T ) . Thus, βj is the proportion of zeros of ξ(s) which are simple, and αj is the proportion which are simple and on the critical line. The best currently available bounds are α0 > 0.40219, α1 > 0.79874, α2 > 0.93469, α3 > 0.9673, α4 > 0.98006, and α5 > 0.9863. These bounds were obtained by combining Theorem 2 of [C2] with the methods of [C1]. Trivially, βj ≥ αj . 1991 Mathematical Subject Classification: 05A20, 11M26 Let Nd(T ) be the number of distinct zeros of ξ(s) in the region 0 < t < T . That is, Nd(T ) = ∞ ∑