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Source. Theorem 3, p. 2 of the arXiv PDF; proof pp. 7--9, using Lemmas 3--6 (pp. 5--6) and the quoted Lemmas 1--2 (p. 2). Read in the text layer and checked on the rendered pages.

Statement

For an integer $k\ge0$ let

$$S_k:=\sum_{n=1}^{\infty}\frac{p_n^k}{n!},$$

$p_n$ the $n$-th prime. Then the real numbers $1,S_0,S_1,S_2,\ldots$ are $\mathbb{Q}$-linearly independent.

In particular every $S_k$ with $k\ge1$ is irrational ($S_0=e-1$). The paper's own framing (p. 2): Erdős stated in 1958 that $S_k$ is irrational and proved $k=1$; "it appears that, for $k>1$, no proof has appeared in print". The statement for all $k\ge1$ therefore has two sources: $k=1$ in Erdős 1958 and $k\ge2$ here.

Reduction (p. 7)

A nontrivial relation $c+\sum_{k\le K}a_kS_k=0$ with rational coefficients has some $a_k\ne0$, and then $S:=\sum_{\nu\ge1}P(p_\nu)/\nu!$ with $P=\sum a_kx^k\ne0$ is rational; clearing denominators, the theorem reduces to the irrationality of $\sum_{\nu\ge1}P(p_\nu)/\nu!$ for every nonzero $P\in\mathbb{Z}[x]$. The paper states this reduction (p. 7) as "It suffices to show that $S=\sum_{\nu=1}^{\infty}P(p_\nu)/\nu!$ is irrational for every polynomial $P$ with integral coefficients which does not vanish identically."

Dependencies

• Lemma 1 (Weyl–van der Corput) and Lemma 2 (Erdős–Turán), p. 2, quoted from the literature ([3, Theorem 2.8], [6, II, Theorem 2.5]): an exponential-sum bound for functions with controlled $(q+2)$-nd derivative, and the discrepancy bound $D_N\ll N/H+\sum_{1\le h\le H}\big|\sum_{n\le N}e(hx_n)\big|$.
• Lemma 3 (p. 5), "consequence of Selberg's sieve, confer, e.g. [4, Theorem 5.1]" (Halberstam–Richert): if $0\le a_1<\cdots<a_k<N$ are integers and $\mathcal{N}\subseteq[x,2x]$ is a set of integers $n$ with every $n+a_i$ prime, then $|\mathcal{N}|\le\frac{c_kx}{\log^{k+1}x}\prod_p(1+\frac1p)^{k+2-\nu(p)}$, $\nu(p)$ the number of distinct residues modulo $p$ among the shifts; in particular $|\mathcal{N}|\ll_kx\log_2^{k+2}x/\log^{k+1}x$, $\log_2$ the iterated logarithm. (The statement writes the shifts as $a_i$ and the residue set as $\{0,\Delta_0,\Delta_0+\Delta_1,\ldots\}$; the range of the product over $p$ is not specified in the statement, and the "in particular" bound is printed without the argument of $\log_2$.)
• Lemma 4 (p. 5): a nonzero $F\in\mathbb{Z}[x_0,\ldots,x_k]$ has $F(\delta_n,\ldots,\delta_{n+k})\ne0$ for almost all $n$, where $\delta_n=p_{n+1}-p_n$.
• Lemma 5 (p. 6): if $P,Q\in k[X_1,\ldots,X_n]$ over a field $k$, $\nu\ne0$ is an integer, and the polynomial $\nu X_1P+P\,Q^{+}-P^{+}Q$ vanishes identically, where $P^{+}=P(X_2,\ldots,X_{n+1})$ and $Q^{+}=Q(X_2,\ldots,X_{n+1})$ are the index shifts, then $P$ vanishes identically. Proof on p. 6 by setting $X_1=0$ and eliminating variables.
• Lemma 6 (p. 6; proof p. 7): for a nonconstant $Q\in\mathbb{Z}[X]$ of degree $d$ with coefficients bounded by $M$, the discrepancy $D$ of the sequence $Q(p_n/n)\bmod1$, $x\le n\le2x$, satisfies $D\ll xe^{-c\sqrt{\log x}}+M^{1/3}x^{2/3}\log^{d/3}x$. The proof replaces $p_n$ by the inverse function of $\mathrm{li}$ at $n$ using the prime number theorem with the classical error term, then applies Lemma 2 and the case $q=0$ of Lemma 1.

Proof structure (pp. 7--9)

Step 1. Assume $S=\sum_{\nu\ge1}P(p_\nu)/\nu!$ is rational, $\deg P=k$. For large $n$, $n!S$ is an integer, so the scaled tail $\sum_{\nu>n}P(p_\nu)/((n+1)\cdots\nu)$ is an integer. Since $p_\nu\sim\nu\log\nu$, only the first few terms matter: the paper writes, with $\|\cdot\|$ the distance to the nearest integer,

$$\left\|\sum_{\nu=1}^{k-1}\frac{P(p_{n+\nu})}{(n+1)\cdots(n+\nu)}\right\| \ll\frac{\log^kn}{n}$$

for large $n$, and calls the truncated sum $F^{(0)}(n)$.

Step 2 (p. 8). Writing $p_{n+i}=p_n+\delta_n+\cdots+\delta_{n+i-1}$ and expanding $1/((n+1)\cdots(n+i))$ in powers of $1/n$ gives

$$F^{(0)}(n)=\sum_{\nu=1}^{k}\sum_{\mu=1}^{\nu} P^{(0)}_{\nu\mu}(\delta_n,\ldots,\delta_{n+k-1})\,\frac{p_n^\nu}{n^\mu}+R(n),$$

where $R(n)$ denotes any error $\ll\log^cn/n$ for almost all $n$ (so that $\delta_nR(n)=R(n)$).

Step 3. Pairs $(\nu,\mu)$ are ordered by $\nu-\mu$, the growth exponent of $p_n^\nu/n^\mu$, then by $\nu$. With $(\nu_0,\mu_0)$ the maximal pair present in $F^{(i)}$, the recursion

$$F^{(i+1)}(n)=P^{(i)}_{\nu_0\mu_0}(\delta_{n+1},\ldots,\delta_{n+k})F^{(i)}(n) -P^{(i)}_{\nu_0\mu_0}(\delta_n,\ldots,\delta_{n+k-1})F^{(i)}(n+1)$$

keeps $\|F^{(i)}(n)\|=R(n)$ and removes the leading monomial; Lemma 5 shows that the new coefficient of $p_n^{\nu_0-1}/n^{\mu_0}$ does not vanish identically. After finitely many steps only pairs with $\mu=\nu$ remain, at least one with a nonzero coefficient.

Step 4. Clearing denominators, there are $\ell$ and polynomials $Q_i\in\mathbb{Z}[X_1,\ldots,X_\ell]$ with $Q_\ell\ne0$ and

$$\left\|\sum_{i=1}^{\ell}Q_i(\delta_n,\ldots,\delta_{n+\ell})\, \frac{p_n^i}{n^i}\right\|=R(n).$$

By Lemma 4 some $Q_i(\delta_n,\ldots,\delta_{n+\ell})\ne0$ for almost all $n$, and for almost all $n$ no $\delta_{n+j}$ exceeds $\log^2n$; so (p. 9, formula (2)) for almost all $n$ there are integers $a_1,\ldots,a_\ell$, not all zero, with $0<|a_i|<\log^An$, such that $\big\|\sum_ia_i(p_n/n)^i\big\|\ll e^{-c\sqrt{\log n}}$.

Step 5. Pigeonhole: one tuple $(a_i)$ serves at least $x/\log^{\ell A}x$ integers $n\le x$, and the paper ends (p. 9): "This clearly contradicts Lemma 6". The count is not written out; it would run through Lemma 6 with $M=\log^Ax$, which bounds the number of $n\in[x,2x]$ with $\|Q(p_n/n)\|\le e^{-c\sqrt{\log x}}$ by $o(x/\log^{\ell A}x)$. $\blacksquare$

Where scrutiny would begin

Recorded for a future review; none has been made. (i) The truncation in step 1 is printed with upper index $k-1$ ($k=\deg P$), but the term of index $\nu$ has size of order $\log^kn/n^{\nu-k}$, so the term of index $k$ is of order $\log^kn$, larger than the stated error $\log^kn/n$, and the truncation must run at least to $k$. (In step 2 the outer index $\nu$ is the power of $p_n$, not the truncation index.) (ii) The bookkeeping of "almost all $n$" through the finitely many recursion steps and the size of the truncation error $R(n)$, including its interaction with the gap bound $\delta_{n+j}\le\log^2n$. (iii) The pigeonhole count against Lemma 6: the constants $A$ and $\ell$ depend on $P$, and $M=\log^Ax$ must keep $M^{1/3}x^{2/3}\log^{\ell/3}x=o(x/\log^{\ell A}x)$, which it does. (iv) In Lemma 3 the constant $c_k$ and the range of the product are not made explicit; Lemma 4 uses only the "in particular" bound.

Bears on. #251 (context: this is the $k\ge2$ half of the theorem the site's remark attributes wholly to Erdős 1958; it says nothing about $\sum p_n/2^n$).