Two Remarks on a Result of Ramachandra

Abstract

Improving on the results of Montgomery [3] and Huxley [1], Ramachandra proved (see Lemma 4 of [5]) the following large value theorem:

THEOREM 1. Let a_n = a_n(N) (n = N+1, . . . , 2N) be complex numbers subject to the condition max |a_n| = O(N^ε) for every ε > 0. Suppose that n N does not exceed a fixed power of T to be defined. Let V be a positive number such that V+1/v= O(T^ε)for every ε > 0. Let S_r (r = 1, 2, ...,R; R≥2) be a set of distinct complex numbers S_r = σ_r + it_r and let min σ_r = σ, 3/4 ≤ σ ≤ 1,

max t_r - min t_r + 20 = T, min |t_r - t_r|≥(log T)².