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Statement

Setting (p. 159). For a sequence $A:a_1<a_2<\cdots$ of positive integers, $f(A:n)=f(n)$ is the number of representations of $n$ as a sum of one or more consecutive terms of $A$, and its average is

$$F(A;x)=F(x)=\frac1x\sum_{n=1}^{x}f(n).\qquad(1)$$

Equation (5) (p. 160). For $A$ the sequence of primes, $a_i=p_i$,

$$F(x)\sim\log2.\qquad(5)$$

Consequence (p. 161). Since the average value of $f(n)$ is $\log2$, the paper concludes that $f(n)=0$ for infinitely many $n$.

Context (pp. 159--160). For contrast the paper recalls LeVeque's results: for $A$ the positive integers, $F(x)=\frac12\log x+\gamma+\frac12\log2-\frac12+O(x^{-1/2})$ (equation (2), p. 159), and for $A$ an arithmetic progression of positive terms with common difference $d$, $F(x)=\frac12\log x+\gamma-\frac12\log\frac d2-\frac12+O(x^{-1/2})$ (equation (4), p. 160); both give $F(x)\sim\frac12\log x$ (equation (3)). The paper states, without giving the argument, that a variation of its method shows (3) holds for every sequence of positive asymptotic density; the density does not enter the leading term.

Source. L. Moser, Notes on Number Theory III: On the sum of consecutive primes, Canad. Math. Bull. 6 (1963), no. 2, 159--161, DOI 10.4153/CMB-1963-013-1. Definitions (1)--(4) on pp. 159--160, equation (5) on p. 160, its proof on pp. 160--161 and the consequence on p. 161. The edition read is identified on the source card.

Read depth. Claims checked: the definitions, equation (5), the consequence and the stated context were read clause by clause on the printed pages; the proof was read for its structure. The paper writes out its final chain of asymptotic estimates but does not justify the steps, saying only that they can easily be justified using (8) and the prime number theorem.

Proof pointer

Pages 160--161. Each block of consecutive primes with sum at most $x$ contributes $1$ to $f(1)+\cdots+f(x)$, and the number of such blocks of $r$ primes lies between $\pi(x/r)-r$ and $\pi(x/r)$ (inequality (6)), for $r$ up to the $k$ with $p_1+\cdots+p_k\le x<p_1+\cdots+p_{k+1}$ (definition (7)). From $p_r\asymp r\log r$ the paper gets $k\asymp\sqrt{x/\log x}$ (estimate (8)), so the error $\sum_{r\le k}r$ is $o(x)$. The prime number theorem then turns $\sum_{r\le k}\pi(x/r)$ into an integral of $x/(r\log(x/r))$ over $1\le r\le k$, equal to $x(\log\log x-\log\log(x/k))$, which is asymptotic to $x\log2$ because $x/k$ is of the order of $\sqrt{x\log x}$.

Dependencies

The prime number theorem and $p_r\asymp r\log r$; no other result of the paper.

Bears on

• Problem 358: the paper's $f(A:n)$ is the problem's $f(n)$ for a sequence of positive integers. Equation (5) and its consequence show that the primes are not an example for either question of the problem: $f(n)=0$ for infinitely many $n$, so neither $f(n)\to\infty$ nor $f(n)\ge2$ for all large $n$ holds for the primes. The paper does not mention Erdős or the problem and says nothing about other sequences beyond the averages recalled above.