K. Ramachandra

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

AbstractWe 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 {an}be a sequence of complex numbers satisfying the inequality

A lattice-point problem for norm forms in several variables

\left| {\sum\limits_{m = 1}^N {a_m - N} } \right| \leqslant \left( {\frac{1}{2} - \theta } \right)^{ - 1}

Effective and noneffective results on certain arithmetical functions

for N = 1,2,3,…,also for n = 1,2,3,…let αnbe real and ¦αn¦ ≤ C(θ)where C(θ) > 0is a certain (small)constant depending only on θ. Then the number of zeros of the function

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

\sum\limits_{n = 1}^N {a_n \left( {n + \alpha _n } \right)^{ - s} } = \zeta \left( s \right) + \sum\limits_{n = 1}^\infty {\left( {a_n \left( {n + \alpha _n } \right)^{ - s} - n^{ - s} } \right)}

A note on recent papers of Ramachandra and Huxley

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 ≥T0(θ,δ)a large positive constant. Theorem B.In the above theorem we can relax the condition on an to