Hancl 2004 irrationality cantor series

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algorithm_3_1: Gives, for a nondecreasing sequence of positive integers b_n with the sum of b_n over 2 to the n convergent, a choice of a_n in {2, 3, 4} making the Cantor series equal any prescribed value in an interval; with b_n the primes, the interval ends at the constant of problem 251.

corollary_4_2: States the exact rationality test for the sum of positive integers b_n over n factorial when b_(n+1) minus b_n is o(n), which reproves the irrationality of the sum of p_n over n factorial from a prime gap bound.

example_2_1: Records the two-line proof that the sums of phi(n) over n factorial and of sigma(n) over n factorial are irrational, from the integrality of the tails at prime indices; the second is the case k equal to one of problem 252.

example_3_1: Proves that the sum of p_n to the k over 2 to the p_n is irrational for every positive integer k, from Theorem 3.1 and Westzynthius's unbounded normalized prime gaps; an adjacent series to that of problem 251.

theorem_3_1: States that positive integers b_n with b_(n+1) below (1 plus epsilon) times b_n for a fixed epsilon below one, and with b_n over a_n tending to zero along a subsequence, give an irrational sum of b_n over a_1 through a_n.

theorem_3_2: States that a Cantor series with a_n never dividing b_n is irrational when the lim inf of |b_n| over a_n is zero and b_n over a_(n-1) a_n tends to zero; the theorem closes one case of the proof of Theorem 5.1.

theorem_5_1: States the exact rationality test for the sum of p_n over a_1 through a_n when a_n is a monotonic sequence of positive integers with p_n of size o(a_n squared), strengthening the 1958 theorem of Erdős and the 1974 theorem of Erdős and Straus.

theorem_5_2: States that monotonic positive integer sequences a_n and b_n with b_n over a_n squared tending to zero, a_(2n) b_(2n) over n a_n squared tending to zero, and gcd(a_n - 1, b_n) small against b_n for a positive proportion of each of infinitely many dyadic ranges give an irrational Cantor series; the growth condition cannot be dropped.

theorem_6_1: States the exact rationality test for the sum of p_n over a_1 through a_n when a_n is a monotonic sequence of positive integers with a_n over log n tending to infinity, the paper's partial affirmation of Erdős's 1958 expectation, with the case a_n equal to two out of reach.

theorem_6_2: States the exact rationality test for the sum of n over a_1 through a_n when a_n is an unbounded monotonic sequence of positive integers; for a bounded monotonic sequence the sum is always rational.

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Jaroslav Hančl and Robert Tijdeman, On the irrationality of Cantor series, J. Reine Angew. Math. 571 (2004), 145--158; Zbl 1049.11076; MSC 11J72.

Edition read

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Contents

Standing convention (p. 3): $\{a_n\}$ and $\{b_n\}$ are rational integers with $a_n>1$ for all $n$; $S=\sum_{n\ge1}b_n/(a_1\ldots a_n)$ (1) and $S_N:=\sum_{n\ge N}b_n/(a_N\ldots a_n)$ (2). Lemma 2.1 (p. 3): if $S=r/q$ then $qS_N\in\mathbb{Z}$ for all $N$; the Remark after it notes that for a rational $S=\sum b_n/n!$ the tails $S_N$ are themselves integers for large $N$. Formula (3) (p. 3): if $b_n=o(a_{n-1}a_n)$ then $|S_n-b_n/a_n|<\epsilon$ for $n\ge n_0(\epsilon)$.

• Example 2.1 (pp. 3--4): $\sum\varphi(n)/n!$ and $\sum\sigma(n)/n!$ are irrational, by Lemma 2.1 and (3) at prime indices.
• Theorem 3.1 (p. 4): $b_n>0$, $b_{n+1}<(1+\epsilon)b_n$ for some $\epsilon<1$ and all $n\ge n_1$, and $\liminf b_n/a_n=0$ imply $S$ irrational; Example 3.1 (p. 4): $\sum p_n^k/2^{p_n}$ is irrational for every integer $k>0$. Proposition 3.1 (with its auxiliary Lemma 3.1), Corollary 3.1, Example 3.2, Corollary 3.2 (Oppenheim [11]) and Theorem 3.2 (pp. 4--6) are further consequences of Lemma 2.1; Theorem 3.2 (p. 5): a series with $a_n\nmid b_n$ for every $n$, $\liminf|b_n|/a_n=0$ and $b_n/(a_{n-1}a_n)\to0$ is irrational.
• Section 4 (pp. 7--8): Proposition 4.1 and Theorem 4.1 (p. 7), Corollary 4.1 and Corollary 4.2 (p. 8): for positive integers $b_n$ with $b_{n+1}-b_n=o(n)$, $\sum b_n/n!$ is rational if and only if $b_n/(n-1)$ is constant for $n\ge n_1$; Example 4.1 (p. 8): $\sum d(n)/n!$ and $\sum(n-d(n))/n!$ are irrational.
• Section 5 (pp. 8--11): Theorem 5.1 (p. 8): for monotonic positive integers $a_n$ with $p_n=o(a_n^2)$, $\sum p_n/(a_1\ldots a_n)$ is rational if and only if $p_n/(a_n-1)$ is constant for $n\ge n_0$; Remark (pp. 9--10) on why monotonicity matters; Theorem 5.2 (p. 10) replaces primality by a gcd condition, with an example (pp. 10--11) showing that its growth condition cannot be dropped; Example 5.1 (p. 11): $\sum(p_n/n!)^k$ is irrational for every integer $k\ge1$.
• Section 6 (pp. 11--12): Theorem 6.1 (p. 11): for monotonic positive integers $a_n$ with $a_n/\log n\to\infty$, $\sum p_n/(a_1\ldots a_n)$ is rational if and only if $p_n/(a_n-1)$ is constant for $n\ge n_0$; Theorem 6.2 (p. 12): for unbounded monotonic positive integers $a_n$, $\sum n/(a_1\ldots a_n)$ is rational if and only if $n/(a_n-1)$ is constant for $n\ge n_0$.
• The abstract also announces that for every monotonic $\{b_n\}$ there is a sequence $a_n\in\{2,3,4\}$ making $S$ rational. Algorithm 3.1 (p. 6) does this for monotonically nondecreasing positive integers $b_n$ with $T=\sum b_n2^{-n}$ convergent: for each $S\in(T/3,T]$ it chooses $a_n\in\{2,3,4\}$ with $\sum b_n/(a_1\ldots a_n)=S$.

The paper on Erdős 1958

The introduction (p. 2) records that Erdős claimed in [3] (1958) the irrationality of $\sum_{n=1}^{\infty}p_n^k/n!$ for every $k=1,2,\ldots$, with $\{p_n\}$ the primes in increasing order, and adds: "Unfortunately he proved only the case $k=1$." Corollary 4.2 is presented as the generalization of that case, and the introduction states it as: "Suppose $\{b_n\}_{n=1}^{\infty}$ is a monotonic sequence with $b_{n+1}-b_n=o(b_n)$. Then $\sum_{n=1}^{\infty}\frac{b_n}{n!}$ is rational if and only if $\frac{b_n}{n-1}$ is constant for $n$ greater than some $n_0$." The same page recalls two earlier results on $\sum p_n/(a_1\ldots a_n)$: Erdős [3] proved it irrational whenever $\{a_n\}$ is monotonically non-decreasing and some $k>0$ has $\lim a_n(\log n)^k/n=\infty$, and Erdős and Straus [7] replaced that growth condition by the pair of conditions $p_n=o(a_n^2)$ and $\liminf a_n/p_n=0$. As recalled there, the first omits the exception in Erdős's theorem, which is an exact test: the sum is rational exactly when $a_n=qp_n+1$ for a fixed integer $q\ge1$ from some index on (statement), as for $a_n=p_n+1$, where it equals $1$. Theorem 5.1 keeps $p_n=o(a_n^2)$ and replaces the second condition by the necessary one, that $p_n/(a_n-1)$ is not constant from some $n_0$ on. Section 6 opens (p. 11) with the authors' assessment, quoted on the Theorem 6.1 page, that Theorem 6.1, which relaxes the growth condition of Theorem 5.1 to $a_n/\log n\to\infty$, partially affirms Erdős's expectation in [3] p. 99 that the monotonicity of $\{a_n\}$ alone suffices, and that the irrationality of $\sum p_n/2^n$ stays out of their reach. Reference [3] is the paper filed as erdos_1958_sur_certaines_series_valeur_irrationnelle_french, [7] is Erdős–Straus 1974, and [4] is the 1980 Erdős–Graham monograph. The introduction's paraphrase of Corollary 4.2 differs from the corollary itself (increments $o(b_n)$ versus $o(n)$), which is why it is quoted exactly; see the corollary's page.

Compiled scope

Statements read on the rendered pages; the short proofs of Example 2.1, Theorem 3.1, Example 3.1, Corollaries 4.1--4.2 and Algorithm 3.1 read in full and sketched on their pages; the proofs of Theorems 3.2, 5.1, 5.2, 6.1 and 6.2 read for structure. None has been independently reviewed.

Bears on. #251 (Corollary 4.2 reproves the irrationality of $\sum p_n/n!$, the 1958 theorem cited on the problem page; Theorems 5.1 and 6.1 are exact rationality tests for $\sum p_n/(a_1\cdots a_n)$ with monotonic $a_n$ under growth conditions that exclude $a_n=2$, and p. 11 records that $\sum p_n/2^n$ stays out of the authors' reach; Algorithm 3.1 with $b_n=p_n$ writes every number of $(T/3,T]$, $T$ the problem's constant, as such a sum with all $a_n$ in $\{2,3,4\}$, and says nothing about $T$ itself; Example 3.1 treats the different series $\sum p_n^k/2^{p_n}$), #252 (Example 2.1 gives the case $k=1$ by an elementary argument).

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