# The Selberg-Delange method in short intervals with some applications

• Zhen Cui
• Guangshi Lü
• Jie Wu
Articles

## Abstract

In this paper, we establish a quite general mean value result of arithmetic functions over short intervals with the Selberg-Delange method and give some applications. In particular, we generalize Selberg's result on the distribution of integers with a given number of prime factors and Deshouillers-Dress-Tenenbaum's arcsin law on divisors to the short interval case.

## Keywords

asymptotic results on arithmetic functions Selberg-Delange method arithmetic functions distribution of integers

11N37

## Notes

### Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11671253, 11771252 and 11531008), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20120073110059), Program for Innovative Research Team in University of Ministry of Education of China (Grant No. IRT16R43) and Taishan Scholars Project, the Program PRC 1457-AuForDiP (CNRS-NSFC). Finally, the authors are grateful to Y-K Lau for his help during the preparation of this paper, and to the referee for a careful reading of our manuscript and helpful suggestions.

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