Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T04:29:35.215Z Has data issue: false hasContentIssue false
  • English
  • Français

On the divisor function d(n): II

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  26 February 2010

Jiahai Kan  and
Zun Shan
Jiahai Kan
Affiliation:
Nanjing Institute of Posts and Telecommunications, 210003 Nanjing, Nanjing, China.
Zun Shan
Affiliation:
Department of Mathematics, Nanjing Normal University, 210097 Nanjing, Nanjing, China.
Get access

Extract

In 1984 Heath-Brown [2] proved the conjecture of Erdős and Mirsky [1], and obtained the following result.

Theorem. There are infinitely many integers n for which d(n+ 1)= d(n). Indeed, for large x, the number of such n≤x is at least of orderx ln-7x.

MSC classification

Secondary: 11K65: Arithmetic functions
Type
Research Article
Information
Mathematika , Volume 46 , Issue 2 , December 1999 , pp. 419 - 423
Copyright
Copyright © University College London 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Erdős, P. and Mirsky, L.. The distribution of the values of the divisor function d(n). Proc. London Math. Soc. (3), 2 (1952), 257–271.CrossRefGoogle Scholar
2. Heath-Brown, D. R.. The divisor function at consecutive integers. Mathematika, 31 (1984). 141–149.CrossRefGoogle Scholar
3. Hildebrand, A.. The divisor function at consecutive integers. Pacific J. Math., 129 (1987), 307–309.CrossRefGoogle Scholar
4. Kan, J. and Shan, Z.. On the divisor function d(n). Mathematika, 43 (1996), 320–322.CrossRefGoogle Scholar