Ein Tauber-Satz mit Restglied für die Laplace-Transformation. (A Tauberian theorem with remainder term for the Laplace transformation) (Q750779)

The classical Wiener-Ikehara Tauberian theorem gives information about the asymptotic behaviour of the [non-decreasing] function \(A(t)\) from knowledge of the Laplace-integral \(f(s)=\int^{\infty}_{0}A(x)\cdot e^{-xs}dx,\) \(s=\sigma +it\), \(\sigma >\alpha\). The author proves a general version of the Ikehara theorem with remainder term: Put \(f_ 0(s)=\int^{\infty}_{0}A_ 0(x)\cdot e^{-xs},\) (Re s\(>\alpha)\), and \(g(s)=f(s)-f_ 0(s).\) Assume \(| A_ 0(y)| \leq B_ 0(y)\), where \(A_ 0\) and \(B_ 0\) satisfy several [reasonable] conditions, which are too complicated to be stated here. Then the difference \(A(x)-A_ 0(y)\) is estimated by \[ | A(y)-A_ 0(y)| \leq const.\{(T/\lambda)B_ 0(y)+\hat h(T)(B_ 0(y)+e^{\sigma y}\int^{\infty}_{- \infty}h(t)| g(\sigma +it)| dt)+ \] \[ +e^{\sigma y}\max_{x=\pm T/\lambda}| \int^{\infty}_{- \infty}h(t/\lambda)e^{it(x+y)}g(\sigma +it)dt| \}, \] where \(h(x)=\exp (-cx^ 2)\), \(\hat h(t)=\exp (-t^ 2/4c)\sqrt{\pi /c}\), \(\alpha <\sigma \leq \alpha +1\), \(T\geq \lambda \geq 1.\) The applicability of this theorem (and of some simpler corollaries) is shown by the following results, remarkable with concern to the good remainder terms, which are deduced from the author's Tauberian theorem (and, of course, using information on the zeta function, on \(\sum \tau^ 2(n)n^{-s}\), where \(\tau\) (n) is Ramanujan's function, on elliptic functions and the generating function of the partition function): (a) \(\sum_{n\leq x}d(n)=x\cdot \log x+(2\gamma -1)x+O(x^{1/3}(\log x)^{7/6})\), (b) \(\sum_{n\leq x}\tau^ 2(n)=x^{12}\{A+O(x^{-2/5}(\log x)^{7/20})\},\) (c) \(\sum_{p^ k\leq x}\log p=x\{1+O(e^{-cw(x)})\},\) where w(x) depends on the zero-free region of \(\zeta\) (s) in the usual manner, (d) \(c(n)=2^{-1/2} n^{-3/4} \exp (4\pi \sqrt{n})\{1+O(\log n/\sqrt{n})\},\) where \(j(\tau)=2^{-6} 3^{-3} e^{-2\pi i\tau}(1+\sum^{\infty}_{1}c(n)e^{2\pi in\tau}),\) (e) \(p(n)=1/4\sqrt{3}n\cdot \exp (\pi \sqrt{2n/3})\{1+O(\log n/\sqrt{n})\}.\)