Hancl 2005 irrationality factorial series

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corollary_3_1: States that the sum of P(N) over N factorial, for an integer polynomial P, is rational exactly when the coefficients of P weighted by Bell numbers sum to zero, and is irrational when the leading coefficient is positive and the others nonnegative.

corollary_3_4: States that for a real polynomial P with nonnegative coefficients and positive leading coefficient the sum of the integer part of P(N) over N factorial is irrational.

corollary_3_5: States that for every real alpha at least zero and every positive real gamma the sum over N of the integer part of gamma N to the alpha divided by N factorial is irrational.

corollary_4_1: States that the numbers 1, e and the sums over n of the integer part of n to the alpha divided by n factorial, for all positive non-integral real alpha, are linearly independent over the rationals.

theorem_3_1: States that for an integer polynomial P the sum of P(N) over the product of an plus b for n up to N is rational exactly when an explicit finite Stirling-type sum of the coefficients of P vanishes.

theorem_3_2: States that if an integer sequence f(N) equals a rational polynomial P(N) plus o(N) and the sum of f(N) over the products of an plus b is rational, then f(N) equals P(N) minus the constant Q_1 of Lemma 3.1.

theorem_3_3: States that if an integer sequence f(N) equals (aN+b)P(N)+O(1) for a real polynomial P and the sum of f(N) over the products of an plus b is rational, then every coefficient of P is rational.

theorem_3_4: States that the sum of f(N) over the products of an plus b is irrational when f(N) equals (aN+b)F(N)+O(1) for a positive function F whose derivatives up to order K satisfy the Taylor, size and limit conditions (18) to (21).

theorem_3_5: States that the sum of f(N) over the products of an plus b is irrational when f(N) equals (aN+b)F(N)+O(1) for a positive function F with K at least one satisfying the Taylor expansion, uniform derivative bound and limit conditions (22) to (25).

theorem_4_1: States that 1, an irrational sum of P(N) over the products of an plus b, and the sums of f(N) over those products for f(N) equal to (aN+b)F(N)+O(1) with F ranging over a family W of smooth functions with separated growth are linearly independent over the rationals.

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Jaroslav Hančl and Robert Tijdeman, On the irrationality of factorial series, Acta Arith. 118 (2005), no. 4, 383--401; doi:10.4064/aa118-4-5; Zbl 1088.11054; MSC 11J72.

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Contents

Notation (p. 3): $a>0$ and $b$ integers with $an+b\ne0$ for all $n\ge1$; $R^*=\sum_{N\ge1}b_N/\prod_{n=1}^N(an+b)$ (1), with $R=\sum b_n/n!$ the case $a=1$, $b=0$; Lemma 2.1: if $R^*=p/q$ then $qR^*_N\in\mathbb{Z}$ for all $N$; Lemma 2.2 (Oppenheim, Theorem 8): if $|b_n|<an+b$ for $n>n_0$ and $\liminf|b_n|/n=0$, then $R\in\mathbb{Q}$ exactly when $b_n$ vanishes for every $n>n_0$; Lemma 2.3: the Stirling numbers of the second kind $S(r,K)=\frac1{K!}\sum_{j=0}^K(-1)^{K-j}\binom Kj j^r$, with $S(r,K)=0$ for $r<K$, $S(K,K)=1$ and $S(r,K)\in\mathbb{N}$ for $r>K>0$.

• Theorem 3.1 (p. 8): for $P\in\mathbb{Z}[x]$, $R^*=\sum_{N\ge1}P(N)/\prod_{n\le N}(an+b)$ is rational if and only if an explicit finite sum $Q_1$ of the coefficients of $P$ vanishes (formula (17)).
• Corollary 3.1 (p. 8): $\sum_{N\ge1}P(N)/N!$ is rational if and only if $\sum_{i=0}^Ta_i\sum_{k=0}^iS(i,k)=0$; if the leading coefficient is positive and the others nonnegative, the sum is irrational.
• Theorem 3.2 (p. 8): for integer numerators $f(N)=P(N)+o(N)$ with $P\in\mathbb{Q}[x]$, a rational sum forces $f(N)=P(N)-Q_1$ (printed "for all $N$"; the proof gives all large $N$).
• Theorem 3.3 (p. 9): numerators $f(N)=(aN+b)P(N)+O(1)$ with $P\in\mathbb{R}[x]$ give a rational sum only if every coefficient of $P$ is rational; hence Corollary 3.4 (p. 10), $\sum[P(N)]/N!\notin\mathbb{Q}$ for real $P$ with nonnegative coefficients and positive leading coefficient.
• Theorem 3.4 (p. 10) and Theorem 3.5 (pp. 11--12): integer numerators $f(N)=(aN+b)F(N)+O(1)$ with $F$ positive and smooth, under the derivative conditions (18)--(21) and (22)--(25) respectively, give an irrational sum whenever it converges absolutely. Corollaries 3.5--3.8 (pp. 11--13) give, for instance, Corollary 3.5, $\sum[\gamma N^\alpha]/N!\notin\mathbb{Q}$ for $\alpha\ge0$, $\gamma>0$, and the series $\sum[\log n]/n!$ and $\sum[\exp(\log^{1/2}n)]/n!$ named on p. 2.
• Section 4 (pp. 14--17): Theorem 4.1 (p. 14) proves linear independence over the rationals of $1$, an irrational polynomial series and series with smooth numerators from a family of separated growth; for example Corollary 4.1 (p. 16): $1$, $e$ and the numbers $\sum[n^\alpha]/n!$ for all $\alpha\in\mathbb{R}^+\setminus\mathbb{Z}$ together (the introduction, p. 2, states it so).

The paper on the prime power factorial series and on problem 252

The introduction (p. 2) places the paper's results as generalizations of Erdős's theorem [3] that $\sum_{n=1}^{\infty}p_n/n!\notin\mathbb{Q}$, with $\{p_n\}$ the primes in increasing order, and records Erdős's claim that $\sum_{n=1}^{\infty}p_n^k/n!$ is irrational for every $k=1,2,\ldots$, adding "but unfortunately he proved only the case $k=1$". The same page surveys the divisor-function series: Oppenheim [12] proved that $\sum\epsilon_nd(n)/n!$, $\sum\epsilon_n\sigma(n)/n!$ and $\sum\epsilon_n\varphi(n)/n!$ are irrational for every choice of signs $\epsilon_n\in\{-1,1\}$, where $d(n)$, $\sigma(n)$ and $\varphi(n)$ are the number of divisors, the sum of divisors and Euler's function of $n$; a special case is due to Erdős and Kac [4]; and Erdős and Straus [5] proved that $1$, $\sum\sigma(n)/n!$, $\sum\varphi(n)/n!$ and $\sum b_n/n!$ are linearly independent over $\mathbb{Q}$ whenever $|b_n|<n^{1/2-\epsilon}$ for all large $n$ and $b_n\ne0$ for infinitely many $n$. The paper notes that most of the results it mentions were stated more generally in the original papers. Reference [3] is Erdős 1958, [4] is Erdős–Kac, Problem 4518, Amer. Math. Monthly 61 (1954), and [5] is Erdős–Straus 1974.

The paper's theorems take numerators that are polynomials, within $o(N)$ of a polynomial, or of the form $(aN+b)F(N)+O(1)$ with $F$ smooth of polynomial growth (p. 2). Neither $p_n^k$ nor $\sigma_k(n)$ is given in any of these forms, and the paper applies no result to them, so no result here bears directly on $\sum p_n^k/n!$ or on $\sum\sigma_k(n)/n!$ for $k\ge2$.

Compiled scope

Statements read on the rendered pages; Theorem 3.1 and Corollary 3.1 with their one-line proofs read in full; Theorems 3.2--3.5 and 4.1 and the corollaries with result pages read clause by clause as statements, their proofs for structure only. No proof is rewritten and none has been independently reviewed.

Bears on. #252 (context: the exact rationality test for factorial series with polynomial numerators, Corollary 3.1, which does not reach $\sigma_k(n)$, and the p. 2 survey of what is known for $\sum\sigma(n)/n!$), #251 (mention: the p. 2 record that Erdős proved the factorial theorem cited on the problem page for $k=1$ only).

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