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Totient quotient and small gaps between primes

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Abstract

For each \(m\ge 1\), there exists \(H=H(m)>0\) such that the set

$$\begin{aligned} \bigg \{\frac{\phi (p+1)}{\phi (p-1)}:\,p\text { is prime and }[p+1,p+H]\text { contains at least }m\text { primes}\bigg \} \end{aligned}$$

is dense in \([0,+\infty )\), where \(\phi \) denotes the Euler totient function. This gives a unconditional weak form of a recent result of Garcia, Luca, Shi and Udell, which was proved under the assumption of a conjecture of Dickson.

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Acknowledgements

The authors are grateful to the anonymous referee for his/her very helpful comments on this paper. The L. Dai is supported by National Natural Science Foundation of China (Grant No. 11571174) and Qing Lan Project of Nanjing Normal University. The H. Pan is supported by National Natural Science Foundation of China Grant No. 12071208.

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Authors and Affiliations

  1. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, People’s Republic of China

    Lixia Dai

  2. School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, 210023, People’s Republic of China

    Hao Pan

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  1. Lixia Dai
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  2. Hao Pan
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Correspondence to Hao Pan.

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Communicated by Alberto Minguez.

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Dai, L., Pan, H. Totient quotient and small gaps between primes. Monatsh Math 194, 481–493 (2021). https://doi.org/10.1007/s00605-020-01502-8

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