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On the order of magnitude of Jacobsthal's function

Published online by Cambridge University Press:  20 January 2009

R. C. Vaughan
Affiliation:
Imperial College, London, S.W.7.
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Let n be an integer with n > 1. Jacobsthal (3) defines g(n) to be the least integer so that amongst any g(n) consecutive integers a + 1, a + 2, … a + g(n) there is at least one coprime with n. In other words, if

then

It is probably true that

where ω(n) denotes the number of different prime divisors of n, and Erdos (1) has pointed out that by the small sieve it is possible to show that there is a constant C such that

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1977

References

REFERENCES

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