Vol. 84, No. 1, 1979

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Generalization of a theorem of Landau

Miriam Hausman

Vol. 84 (1979), No. 1, 91–95
Abstract

A well known theorem of Landau asserts that

     ϕ(n-)loglogn-   −γ
nli→m∞     n      = e
(1.1)

where γ = Euler’s constant. In this paper a generalization is obtained by focusing on

                   1∕k     ϕ-(n-+-1)     ϕ(n-+-k)-
G (k) = lnim→∞ (loglogn)   max ( n+ 1  ,⋅⋅⋅ , n+ 1  ).
(1.2)

Clearly, the assertion G(1) = eγ is precisely Landau’s theorem. It is proved that

           ∏
G(k) = e−γ∕k  (1− 1)−1∕kψ(k)
p<k    p
(1.3)

where

      ∏           ∏
ψ(k) =   (1− 1)1∕p    (1 − 1)(1∕k)[k∕p]+1∕k.
p|k    p    p∤k     p
p<k         p<k
(1.4)

Mathematical Subject Classification
Primary: 10A20, 10A20
Milestones
Received: 27 January 1977
Revised: 8 March 1979
Published: 1 September 1979
Authors
Miriam Hausman