A Tauberian theorem and analogues of the prime number theorem

In 1945 Ingham (3) proved the following Tauberian theorem: if ƒ is a non-decreasing, non-negative function on [1, ∞) and 1 then ƒ(x) ∼ cx . His proof is based on the non-vanishing of the Riemann zeta-function, ζ (s) , on the line , and uses Pitt's form of Wiener's Tauberian theorem; (see, e.g., 5, Theorem 109, p. 211).