Schlagepuchta 2011 irrationality number theoretical series

../

lemma_4: States that a nonzero integer polynomial evaluated at k plus one consecutive prime gaps is nonzero for almost all n, proved from a Selberg-sieve bound on shifted prime tuples.

theorem_1: Proves that one, e and the series of the integer parts of n to the lambda over n factorial, for all non-integral positive lambda, are linearly independent over the rationals, with Proposition 1 on the map from lambda to that series.

theorem_2: Proves that if, for a base b that is not a proper power, the base-b concatenation of the digits of a nondecreasing sequence f(n) with regular ratios is rational, then the ratios converge to a power of b and f(n plus one) equals that power times f(n) up to a bounded error; unrelated to problem 251 except by analogy.

theorem_3: Proves that one and the series S_k of p_n to the k over n factorial, for all k at least zero, are linearly independent over the rationals, which gives the irrationality of each S_k for k at least two, a case Erdős stated and the paper says had no proof in print.

---

Jan-Christoph Schlage-Puchta, The irrationality of some number theoretical series, Acta Arithmetica 126 (2007), no. 4, 295--303; doi:10.4064/aa126-4-1; Zbl 1107.11030; MSC 11J72. Also arXiv:1105.1451 [math.NT], posted 7 May 2011.

Identity and edition

The source is the refereed 2007 Acta Arithmetica article; the arXiv record of 2011 is an alias of it, and the slug year records that posting. The copy read for this card is the arXiv version: nine pages, header "arXiv:1105.1451v1 [math.NT] 7 May 2011", running head "THE IRRATIONALITY OF SOME NUMBER THEORETICAL SERIES", author's address Mathematisches Institut, Freiburg. Provenance: the file served at https://arxiv.org/pdf/1105.1451v1 on 2026-09-17 (UTC), 135,121 bytes. The journal version was not compared; page numbers and labels below are those of the arXiv PDF (pp. 1--9). Its text layer is reliable; the statements were also checked on the rendered pages 1--9. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1105.1451), every other right reserved.

Not the same paper as schlagepuchta_2006_irrationality_number_theoretical_series (Ramanujan J. 12 (2006), 455--460, on $\sum\sigma_k(n)/n!$, problem 252), whose title differs by one article.

Contents

The paper proves the irrationality of several series "by combining methods from elementary and analytic number theory with methods from the theory of uniform distribution" (p. 1).

• Theorem 1 (p. 1; proof pp. 2--3): for real $\lambda\ge0$ put $S_\lambda=\sum_{n\ge0}[n^\lambda]/n!$; the set $\{1,e\}\cup\{S_\lambda:\lambda\in(0,\infty)\setminus\mathbb{Z}\}$ is $\mathbb{Q}$-linearly independent. Proposition 1 (p. 1; proof pp. 3--4) describes the map $\lambda\mapsto S_\lambda$ (injective, monotone, continuous from the right; its image has the cardinality of the continuum, Hausdorff dimension $0$, and is totally disconnected). The proof of Theorem 1 uses the equidistribution of $f(n)$ modulo $1$ via Lemma 1.
• Theorem 2 (p. 1; proof pp. 4--5): for a base $b$ that is not a proper power, a rational number whose base-$b$ digits are the concatenated digits of a regularly growing sequence $f(n)$ forces $f(n+1)=cf(n)+O(1)$ with $c$ a power of $b$.
• Theorem 3 (p. 2; proof pp. 7--9): with $S_k=\sum_{n\ge1}p_n^k/n!$, the real numbers $1,S_0,S_1,S_2,\ldots$ are $\mathbb{Q}$-linearly independent. Its tools are Lemma 3 (p. 5, a Selberg-sieve bound cited to Halberstam–Richert), Lemma 4 (p. 5, a nonzero polynomial in consecutive prime gaps vanishes for few $n$), Lemma 5 (p. 6, an algebraic nonvanishing lemma) and Lemma 6 (p. 6, a discrepancy bound for $Q(p_n/n)$ modulo $1$), the last built from the quoted Lemma 1 (Weyl–van der Corput) and Lemma 2 (Erdős–Turán) on p. 2.

The sentence preceding Theorem 3 (p. 2), verbatim: "P. Erdös[2] stated that $S_k$ is irrational and gave a proof for $k=1$. However, it appears that, for $k>1$, no proof has appeared in print. Our last result is the following." Reference [2] is the 1958 Enseignement Math. paper filed as erdos_1958_sur_certaines_series_valeur_irrationnelle_french.

Standing

Theorem 3 proves the irrationality of $\sum p_n^k/n!$ for $k\ge2$, which Erdős stated in 1958 and proved only for $k=1$; the paper's own account (p. 2) is that "it appears that, for $k>1$, no proof has appeared in print", and this card makes no priority claim beyond that. Its acceptance rests on the refereed publication and on the citation in Hančl–Tijdeman 2010 (p. 2: "Erdős's claim that $\sum p_n^k/n!\notin\mathbb{Q}$ is irrational for $k=2,3,\ldots$ was recently confirmed by Schlage-Puchta"). No independent review of the proof has been made here; the result pages record statements, proof structure and the points where scrutiny would begin. No formalization of Theorem 3 is known to this library.

Bears on. #251: Theorem 3 proves the cases $k\ge2$ of the statement that the remark on erdosproblems.com/251 attributes to Erdős 1958 ($\sum p_n^k/n!$ irrational for every $k\ge1$), whose case $k=1$ Erdős proved; it is not progress on the problem's own series $\sum p_n/2^n$, and Theorem 2 is related to that series by analogy only. Theorem 1 and Lemma 4 bear on no catalog problem directly.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.