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Statement

Setting (p. 1): for a natural number $k$, $\sigma_k(n)=\sum_{d\mid n}d^k$ is the sum of the $k$th powers of the divisors of $n$, and

$$S_k=\sum_{n\ge1}\frac{\sigma_k(n)}{n!}.$$

Theorem (p. 1, unnumbered, quoted). "Define $S_k$ as above. (1) If Schinzel's conjecture H is true, then $S_k$ is irrational for all $k\in\mathbb N$. (2) $S_3$ is irrational."

Part (2) carries no hypothesis.

Schinzel's conjecture H as the paper states it (p. 1, quoted). "Let $P_1,\dots,P_k$ be integral polynomials with positive leading coeficients [sic], such that for each prime number $p$ there exists some integer $a$ such that $P_1(a)\cdots P_k(a)\not\equiv0\pmod p$. Then there exist infinitely many integers $n$ such that $P_i(n)$ is prime for $1\le i\le k$." The paper cites Schinzel and Sierpiński (Acta Arith. 4 (1958), 185--208) for it. The printed wording does not require the $P_i$ to be irreducible, which the usual formulation of Hypothesis H does; the proof applies it only to linear polynomials, which are irreducible (an observation of this page).

Earlier cases named by the paper (p. 1): for $k=0,1$ the irrationality of $S_k$ follows from a general result of Erdős and Straus (Pacific J. Math. 55 (1974), 85--92), and for $k=2$ it was shown by Erdős and Kac (Amer. Math. Monthly 61 (1954), Problem 4518). The paper attributes the question whether $S_k$ is irrational for all $k$ to Erdős (New advances in transcendence theory, Cambridge Univ. Press, 1988, 102--109).

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. Page numbers are those of that posting (pp. 1--5); the journal pagination was not compared. The theorem and Hypothesis H on p. 1, the proof of part (1) on pp. 1--2, the proof of part (2) on pp. 2--5.

Read depth. Claims checked: the theorem, the definitions of $\sigma_k$ and $S_k$ and the statement of Hypothesis H were read clause by clause on the page images. The proof was read for the outline below but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Part (1), pp. 1--2. If $S_k=a/b$, then for $n>b$ the number $(n-1)!\,S_k$ is an integer, so the tail $\sum_{\nu\ge n}\sigma_k(\nu)/(\nu)_{\nu-n+1}$, with $(x)_m=x(x-1)\cdots(x-m+1)$, is an integer, and its first $k$ terms lie within $n^{-1+\epsilon}$ of an integer. Hypothesis H supplies primes $q\equiv1\pmod{k!^k}$ for which $(q+i)/(i+1)$ is prime for every $i\le k$; for these the first $k$ terms are explicit up to $O(q^{-1})$. Running the same argument with $q=pr$ for a fixed prime $p>k$ and $r$ prime, and comparing the two estimates, gives $\|q_1^{k-1}/p^k\|<q_1^{-1+\epsilon}$ for arbitrarily large $q_1$. For $q_1>p^2$ the left side is a nonzero rational with denominator dividing $p^k$, so it is at least $p^{-k}$, a contradiction.

Part (2), pp. 2--5. Hypothesis H is replaced by the sieve Lemma (p. 2): at least of order $x/\log^3x$ primes $q\le x$ have $(q+1)/2$ and $(q+2)/3$ free of prime factors up to $x^{1/9}$. For such $q$ the first three terms of the tail fix $9\sigma_3(n)/(4n^2)$, with $n=(q+1)/2$, to within $O(n^{-1/3})$ of a constant modulo 1, for at least of order $x/\log^3x$ integers $n\le x$. An upper count of such $n$, split by the number of prime factors of $n$ and using the Erdős--Turán inequality with van der Corput estimates, gives $O(x\log\log x/\log^4x)$, a contradiction.

The paper prints the fractional part on p. 2 as $7/8+O(q^{-1/3})$ and the resulting constant on p. 3 as $19/216$. Since $\sigma_3(q+1)$ is $\frac98(q+1)^3$ up to a relative error $O(q^{-1/3})$ for these $q$, the fractional part of $\sigma_3(q+1)/(q(q+1)^2)$ is $1/8+O(q^{-1/3})$, and the constant becomes $35/216$. The upper count uses the constant only in the case of one prime factor, where it needs the constant plus or minus $1/4$ to be a non-integer, which holds for $35/216$ as for $19/216$ (an observation of this page, not of the paper).

Dependencies

• Lemma (p. 2), for part (2), which the paper derives from Halberstam and Richert, Sieve methods (1974), Theorem 7.4.
• Part (1) assumes Schinzel's Hypothesis H, which is unproved.

Bears on

• Problem 252: the problem asks, for each $k\ge1$, whether $\sum_{n\ge1}\sigma_k(n)/n!$ is irrational. Part (2) answers yes for $k=3$, unconditionally, and says nothing unconditional about any other $k$. Part (1) answers yes for every $k$ only under Hypothesis H. The problem's claim pages record them separately: the claim page for part (2) and the conditional claim page for part (1).