Generalization of a theorem of Landau

Hausman, Miriam

Recent Issues

Vol. 332: 1 2

Vol. 331: 1 2

Vol. 330: 1 2

Vol. 329: 1 2

Vol. 328: 1 2

Vol. 327: 1 2

Vol. 326: 1 2

Vol. 325: 1 2

Online Archive

Volume:

Issue:

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Officers

Subscriptions

ISSN 1945-5844 (electronic)

ISSN 0030-8730 (print)

Special Issues

Author index

To appear

Other MSP journals

Generalization of a theorem of Landau

Miriam Hausman

Vol. 84 (1979), No. 1, 91–95

DOI: 10.2140/pjm.1979.84.91

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) = e^−γ 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

Vol. 84, No. 1, 1979

Miriam Hausman

Abstract

Mathematical Subject Classification

Milestones

Authors