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On Highly Composite Numbers
Author(s) -
Erdös P.
Publication year - 1944
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/19.75_part_3.130
Subject(s) - composite number , composite material , materials science
for a certain c. In fact I shall prove that if n is highly composite, then the next highly composite number is less than n+n(log y&)-C ; and the result just stated follows immediately from this. At, present I cannot, decide whether the number of highly composite numbers not exceeding z is greater than (logx)” for every k. The principal tool in the proof will be Ingham’s improvement,$ on Hoheisel’s theorem. This asserts that if x is sufficiently large, then the number of primes in the int,erval (x, x+&) is asymptot.ic to cxg(logz)-1. First we state three lemmas, which will be proved at the end of the paper. They are contained subst’antially in the paper of Ramanujan, but we prove Ohem here for completeness. Let n = 22 3~ . . . p+ be a sufficiently large highly composite number. Plainly