Abstract
One of the most interesting results of Hardy and Littlewood was their Tauberian theorem for Lambert summability [1921]; see Section I.10, or Theorem 12.1 below. This result implies the prime number theorem (PNT), but the proof made use of estimates from number theory somewhat stronger than the PNT! There was another Tauberian-type theorem which implies the PNT and whose state was unsatisfactory. That was Landau’s complex Tauberian theorem [1908], [1909] for Dirichlet series, which required more information about the Riemann zeta function in the half-plane {Re z ≥ 1} than seemed reasonable (see Section III.2).
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© 2004 Springer-Verlag Berlin Heidelberg
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Korevaar, J. (2004). Wiener’s Theory. In: Tauberian Theory. Grundlehren der mathematischen Wissenschaften, vol 329. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-10225-1_2
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DOI: https://doi.org/10.1007/978-3-662-10225-1_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05919-3
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