---

Source. Theorem 1, printed p. 213, physical PDF p. 2; Lemmas 2 and 3 on pp. 214--215; the proof of the theorem on p. 215. Read on the page images.

Statement

For every integer $t>1$ (the paper's standing convention; the theorem does not restate the base), the series

$$\sum_{n=1}^{\infty}\frac{1}{t^{\varphi(n)}} \qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{1}{t^{\sigma(n)}}$$

are irrational, where $\varphi$ is Euler's function and $\sigma$ the sum of divisors.

Proof structure (pp. 213--215)

The theorem is Lemma 1 applied to value-counting coefficients. Write $a_k$ for the number of $l$ with $\varphi(l)=k$ and $a'_k$ for the number with $\sigma(l)=k$, so that

$$\sum_{n=1}^{\infty}\frac{1}{t^{\varphi(n)}}=\sum_{k=1}^{\infty}\frac{a_k}{t^k}, \qquad \sum_{n=1}^{\infty}\frac{1}{t^{\sigma(n)}}=\sum_{k=1}^{\infty}\frac{a'_k}{t^k}.$$

• Lemma 2 (p. 214): for a constant $c$, fewer than $cx$ integers $n$ satisfy $\varphi(n)\le x$, and likewise for $\sigma$ (trivially, since $\sigma(n)\ge n$). Proof: $\sum_{m\le x}(m/\varphi(m))^2\le \sum_{m\le x}\prod_{p\mid m}(1+6/p)\le c_1x$, so fewer than $c_1x/r^2$ integers $m<x$ have $m/\varphi(m)>r$; an integer $m>x$ with $\varphi(m)<x$ lies in some range $2^kx<m<2^{k+1}x$ with $m/\varphi(m)>2^k$, and summing the resulting bounds $c_1x/2^{k-1}$ (3) over $k$ gives fewer than $cx$ such $m$. Footnote 2 records the sharper $cx+o(x)$ count due to Erdős and Turán.
• Lemma 3 (pp. 214--215): as $x\to\infty$, only $o(x)$ integers $n\le x$ are values of $\varphi$, and only $o(x)$ are values of $\sigma$ (footnote 3 attributes the $\varphi$ case to S. S. Pillai and points to sharper results). Proof for $\varphi$: fix $r$ with $2^r>2/\varepsilon$; if $k$ has at least $r$ distinct prime factors then $2^r\mid\varphi(k)$, so these $k$ give fewer than $x/2^r<\varepsilon x/2$ values below $x$; if $k$ has fewer than $r$ prime factors then $\varphi(k)>k/r$, so $k<rx$, and Landau's bound (4) on the number of integers up to $y$ with fewer than $r$ prime factors, $cy(\log\log y)^{r-1}/((r-1)!\log y)$, makes these $o(x)$. For $\sigma$: $k\le x$; write $k=a^2b$ with $b$ squarefree; if $b$ has at least $r$ prime factors then $2^r\mid\sigma(k)$; the $k$ with $a>4/\varepsilon$ number at most $\varepsilon x/4$; the remaining $k$, with $b$ having fewer than $r$ prime factors, are $o(x)$ by (4).
• Theorem 1 (p. 215): Lemma 2 gives the bounded-mean condition (2) for $a_k$ and $a'_k$; Lemma 3 gives $f(n)/n\to0$ for both supports; $f(n)\to\infty$ since $\varphi$ and $\sigma$ take infinitely many values; hence Lemma 1 applies to both series.

The closing remark (p. 215) says the same irrationality is clear for the wider family of multiplicative functions treated by Kanold (J. Reine Angew. Math. 195 (1955), 180--195); the author expects it for a far larger class of multiplicative functions but had not proved it.

These steps were read for structure and are recorded as a sketch; no complete rewritten proof and no independent review exist here.

Relation to Problems 249 and 250

These are the exponent variants: $\varphi(n)$ and $\sigma(n)$ sit in the exponent of $t$. Problems 249 and 250 ask about $\sum\varphi(n)/2^n$ and $\sum\sigma(n)/2^n$, where the arithmetic function is the coefficient, and the theorem says nothing about them; the same paper states on p. 212 that those series could not be proved irrational (see the remark on p. 212). The sentence "It is not too hard to prove that $\sum_n\frac{1}{2^{\phi(n)}}$ and $\sum_n\frac{1}{2^{\sigma(n)}}$ are irrational" on printed p. 61 of the 1980 Erdős–Graham monograph refers to this theorem.

Bears on. #249 and #250, as an adjacent exponent variant only; it is not progress on either problem.