Germain prime conjecture

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Let x≥1x\geq 1 be large, and let Λ\Lambda be the von Mangoldt function. Germain prime conjecture.

∑n≤xΛ(n)Λ(2n+1)=s1x+O(xlog⁡x),\sum_{n\leq x}\Lambda(n)\Lambda(2n+1)=s_1x+O\left(\frac{x}{\log x}\right),

where

s1=∏p≥3(p(p−1)(p−1)2)=0.6601618158468….s_1=\prod_{p\geq 3}\left(\frac{p(p-1)}{(p-1)^2}\right)=0.6601618158468\ldots.

This predicts a quantitative distribution for Sophie Germain primes, namely primes pp for which 2p+12p+1 is also prime; the asserted asymptotic is open.

References

Primary source

N. A. Carella, “Primes In Fractional Sequences”, arXiv:1809.02821 (2019).

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