# denotes the i th prime number. The number of the positive divisors of n ∈ N is denoted by d ( n ) , and we write

(1)

## On Large Values of the Divisor Function

P. ERD ˝OS

J.-L. NICOLAS jlnicola@in2p3.fr

Institut Girard Desargues, UPRES-A-5028, Math´ematiques, Bat. 101, Universit´e Claude Bernard (LYON 1), F-69622 Villeurbanne c´edex, France

A. S ´ARK ¨OZY sarkozy@cs.elte.hu

E¨otv¨os Lor´and University, Dept. of Algebra and Number Theory, H-1088 Budapest, M´uzeum krt. 6-8, Hungary

Jean-Louis Nicolas and Andr`as S´ark¨ozy dedicate this paper to the memory of Paul Erd˝os Received June 10, 1997; Accepted January 7, 1998

Abstract. Let d(n)denote the divisor function, and let D(X)denote the maximal value of d(n)for nX . For 0<z1, both lower and upper bounds are given for the number of integers n with nX,z D(X)d(n).

Key words: division function, highly composite numbers, maximal order 1991 Mathematics Subject Classification: Primary 11N56

i

nX

1

2

1

2

3

4

5

k

k(X)

k(X)

### .

Research partially supported by Hungarian National Foundation for Scientific Research, Grant No. T017433 and by C.N.R.S, Institut Girard Desargues, UPRES-A-5028.

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## References

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