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The Ramanujan Journal

, Volume 47, Issue 2, pp 427–433 | Cite as

On the sum of the reciprocals of the differences between consecutive primes

  • Nian Hong Zhou
Article
  • 121 Downloads

Abstract

Let \(p_n\) denote the n-th prime number, and let \(d_n=p_{n+1}-p_{n}\). Under the Hardy–Littlewood prime-pair conjecture, we prove
$$\begin{aligned} \sum _{n\le X}\frac{\log ^{\alpha }d_n}{d_n}\sim {\left\{ \begin{array}{ll} \quad \frac{X\log \log \log X}{\log X}~\qquad \quad ~ &{}\alpha =-1,\\ \frac{X}{\log X}\frac{(\log \log X)^{1+\alpha }}{1+\alpha }\qquad &{}\alpha >-1, \end{array}\right. } \end{aligned}$$
and establish asymptotic properties for some series of \(d_n\) without the Hardy–Littlewood prime-pair conjecture.

Keywords

Differences between consecutive primes Hardy–Littlewood prime-pair conjecture Applications of sieve methods 

Mathematics Subject Classification

Primary 11N05 Secondary 11N36 11A41 

Notes

Acknowledgements

The author would like to thank the anonymous referees and the editors for their very helpful comments and suggestions. The author also thank Min-Jie Luo for offering many useful suggestions and help.

References

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsEast China Normal UniversityShanghaiPeople’s Republic of China

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