On the difference in values of the Euler totient function near prime arguments

Garcia, Stephan Ramon; Luca, Florian

doi:10.1007/978-3-319-92777-0_4

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On the difference in values of the Euler totient function near prime arguments

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Luca, Florian
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/978-3-319-92777-0_4
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arXiv:1706.00392.pdf
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Citation

Garcia, S. R., & Luca, F. (2018). On the difference in values of the Euler totient function near prime arguments. In Irregularities in the distribution of prime numbers: from the era of Helmut Maier’s matrix method and beyond (pp. 69-96). Cham: Springer.

Cite as: https://hdl.handle.net/21.11116/0000-0003-DE41-A

Abstract

We prove unconditionally that for each $\ell \geq 1$, the difference
$\phi(p-\ell) - \phi(p+\ell)$ is positive for $50\%$ of odd primes $p$ and negative for $50\%$.