On the zeros of a class of generalised Dirichlet series-XIV

We prove a general theorem on the zeros of a class of generalised Dirichlet series. We quote the following results as samples. Theorem A.Let 0<θ<1/2and let {a _n }be a sequence of complex numbers satisfying the inequality for N = 1,2,3,…,also for n = 1,2,3,…let α _n be real and |α_n| ≤ C(θ)where C(θ) > 0is a certain (small)constant depending only on θ. Then the number of zeros of the function in the rectangle (1/2-δ⩽σ⩽1/2+δ,T⩽t⩽2T) (where 0<δ<1/2)is ≥C(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided T ≥T ₀(θ,δ)a large positive constant. Theorem B.In the above theorem we can relax the condition on a _n to and |a_N| ≤ (1/2-θ)^-1.Then the lower bound for the number of zeros in (σ⩾1/3−δ,T⩽t⩽2T)is > C(θ,δ) Tlog T(log logT)^-1.The upper bound for the number of zeros in σ⩾1/3+δ,T⩽t⩽2T) isO(T)provided for every ε > 0. Dedicated to the memory of Professor K G Ramanathan