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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.
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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

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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