In a series of papers, the Soviet mathematician I.M. Vinogradov developed a very important method of dealing with estimation of trigonometric sums. (See Chapter VI of [ECT] and [ECT, DRHB]).
In a recent paper [2] R. Balasubramanian and K. Ramachandra proved results like max |ζ(1/2+it)|>t0-δ where ∈ is an arbitrary positive constant, t0 exceeds a positive constant depending on ∈ and C(∈) depends on ∈. In fact their results were very general and they could replace ζ(1/2+it) by F(σ+it) for very general Dirichlet series P(s), and prove (1) for F(σ+it). In this paper we record three theorems and indicate their proof. These are probably well-known to the experts in this field or at least within their easy reach. But the results are so interesting that they deserve to be printed.
