Ingham Tauberian Theorem with an Estimate for the Error Term

We estimate the error term in the Ingham Tauberian theorem. This estimation of the error term is accomplished by considering an elementary proof of a weak form of Wiener's general Tauberian theorem and by using a zero-free region for the Riemann zeta function. 1. Introduction. As an important application of his general Tauberian theorem (GTT), in 1932, Wiener [6] gave a new proof of the prime number theorem (PNT). In 1945, Ingham [2] applied Wiener's GTT to formulate a new Tauberian theorem (now bearing his name) and deduced the PNT as a special case. In 1964, Levinson [3] rediscovered Ingham's Tauberian theorem with a different proof. On the other hand, in 1973, Levinson [4] did show that a weak formulation of Wiener's GTT is enough for the proof of the PNT. In 1981, Balog [1] formulated Ingham's Tauberian theorem with an estimate for the error term. In this paper, we use Levinson's approach to Ingham's theorem, as well as Levinson's approach to Wiener's GTT, to prove the following effective version of Ingham's Tauberian theorem.