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Statement

Notation (p. 2): $P^-(n)$ is the least prime factor of $n$.

Lemma (p. 2, unnumbered, quoted). "The number of primes $p\le x$ such that $P^-\bigl(\frac{p+1}2\bigr)$ and $P^-\bigl(\frac{p+2}3\bigr)$ are both greater then [sic] $x^{1/9}$ is $\gg\frac{x}{\log^3x}$."

The lemma leaves implicit that $(p+1)/2$ and $(p+2)/3$ are integers, that is $p\equiv1\pmod 6$ (an observation of this page). The paper adds (p. 2) that the exponent $1/9$ is not optimal but suffices for its purpose.

Source. J.-C. Schlage-Puchta, The irrationality of a number theoretical series, Ramanujan J. 12 (2006), no. 3, 455--460, doi:10.1007/s11139-006-0154-3, read in the arXiv posting arXiv:1105.1452v1 (7 May 2011) identified on the source card: the lemma on p. 2 of that posting; the journal pagination was not compared.

Read depth. Claims checked: the statement was read clause by clause on the page image. The cited theorem of Halberstam and Richert was not consulted, so its derivation is unchecked here. Nothing here is independently reviewed.

Proof pointer

P. 2: the paper's proof is the single sentence that the lemma follows from Halberstam and Richert, Sieve methods, London Math. Soc. Monographs 4 (1974), Theorem 7.4. No argument is written out.

Dependencies

Halberstam and Richert, Sieve methods (1974), Theorem 7.4, outside this wiki.

Bears on

• Problem 252: the lemma is the sieve input to part (2) of the paper's Theorem, the irrationality of $\sum_{n\ge1}\sigma_3(n)/n!$, standing in for Hypothesis H. By itself it says nothing about the series.