The paper deals with a renewal process whose distribution of the distances between jumps has a density f such that \(\int^{\infty}_{0}x^ af(x)dx<\infty\) for some \(a\geq 1\). With the help of a Tauberian theorem for Laplace transforms due to the author, asymptotic formulas as \(x\to \infty\) for the renewal function of the process are established. These formulas have remainder terms of the kind \(o(x^{-2a+3})\) if \(1\leq a<3\), and \(O(x^{-a}\log x)\) if \(a\geq 3\).