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On convolutions with the Möbius function

Author: S. L. Segal
Journal: Proc. Amer. Math. Soc. 34 (1972), 365-372
MSC: Primary 10K20
Erratum: Proc. Amer. Math. Soc. 39 (1973), 652.
MathSciNet review: 0299572
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Abstract: By using the results of [6], it is proved that for an extensive class of increasing functions h,

$\displaystyle \sum\limits_{1 \leqq d \leqq x} {\frac{{\mu (d)}}{d}h\left( {\frac{x}{d}} \right)} \sim xh'(x)\quad {\text{as}}\;x \to \infty$ ($ *$)

where $ \mu $ denotes the Möbius function. This result incidentally settles affirmatively Remark (iii) of [6], and refines the Tauberian Theorem 2 of that paper. It is also shown that one type of condition imposed in [6] is necessary to the truth of the cited Theorem 2, at least if some sort of quasi-Riemann hypothesis is true. Nevertheless, examples are given to show that on the one hand $ ( ^\ast )$ may be true for functions not covered by the first theorem of this paper, and on the other that some sort of nonnaïve condition on a function h is necessary to ensure the truth of $ (^\ast)$.

References [Enhancements On Off] (What's this?)

  • [1] W. E. Briggs and S. Chowla, The power series coefficients of 𝜁(𝑠), Amer. Math. Monthly 62 (1955), 323–325. MR 0069209
  • [2] G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
  • [3] A. E. Ingham, Some Tauberian theorems connected with the prime number theorem, J. London Math. Soc. 20 (1945), 171–180. MR 0017392
  • [4] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 2nd ed., Chelsea, New York, 1953. MR 16, 904.
  • [5] J. E. Littlewood, Quelques conséquences de l'hypothèse que la fonction $ \zeta (s)$ de Riemann n'a pas de zéros dans le demi-plan $ R(s) > \tfrac{1}{2}$, Comptes Rendus 154 (1912), 263-266.
  • [6] Sanford L. Segal, A general Tauberian theorem of Landau-Ingham type, Math. Z. 111 (1969), 159–167. MR 0249379
  • [7] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485
  • [8] J. R. Wilton, A note on the coefficients of the expansion of $ \zeta (s,x)$ in powers of $ s - 1$, Quart. J. Math. 50 (1927), 329-332.

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Keywords: Möbius function, Dirichlet convolution, Tauberian theorems, Riemann zeta-function, Ingham summability
Article copyright: © Copyright 1972 American Mathematical Society