Moser 1963 notes number theory

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equation_5: Moser proves that when f(n) counts the representations of n as a sum of one or more consecutive primes, the average of f(1), ..., f(x) is asymptotic to log 2, so f(n) = 0 for infinitely many n.

problems_p161: Moser's four closing problems ask, for f(n) the number of representations of n as a sum of consecutive primes, whether f(n) = 1 infinitely often, whether every value k is taken, whether each level set has a density, and whether the upper limit of f(n) is infinite; the paper leaves them open.

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Moser, L., Notes on number theory. III. On the sum of consecutive primes. Canad. Math. Bull. 6 (1963), no. 2, 159-161. No notice is printed in the copy read beyond its "Published online" footer; the journal's article page (https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/on-the-sum-of-consecutive-primes/99CB1E53FB57B2D2084D58D2383B1905) states "Copyright © Canadian Mathematical Society 1963", every other right reserved.

Writing f(n) for the number of ways to write n as a sum of one or more consecutive primes and F(x) for the average of f over 1..x, Moser proves F(x) -> log 2 (his equation (5)). The proof counts blocks of r consecutive primes with sum at most x, which is between pi(x/r) - r and pi(x/r), sums this over r up to the largest admissible block length k determined by p_1 + ... + p_k <= x, and evaluates the resulting sum of pi(x/r) by the prime number theorem as x(log log x - log log sqrt(x log x)) ~ x log 2. He contrasts this with LeVeque's earlier results for the integers and for arithmetic progressions, where F(x) ~ (1/2) log x, and states without proof that a variation of his argument gives the same (1/2) log x behavior for any sequence of positive asymptotic density. Since the mean of f is log 2 < 1, f(n) = 0 for infinitely many n; he then poses four questions - whether f(n) = 1 infinitely often, whether f(n) = k is solvable for every k, whether the n with f(n) = k have a density, and whether lim sup f(n) is infinite. Problem 358 asks whether some sequence A has f(n) -> infinity, or f(n) >= 2 for all large n; this note sets up the representation function f(A:n) for a general sequence and supplies the case of the primes, with the log 2 average and the open questions above.

Source: https://doi.org/10.4153/CMB-1963-013-1.

Results. Labels and pages are those of the print.

• Equation (5) (p. 160, proof pp. 160-161): for A the primes, F(x) ~ log 2, and hence f(n) = 0 for infinitely many n (p. 161). The page also records LeVeque's results (2)-(4) as the paper recalls them and the paper's unproved remark that F(x) ~ (1/2) log x for every sequence of positive asymptotic density.
• Problems 1-4 (p. 161): for the primes, whether f(n) = 1 infinitely often, whether f(n) = k is solvable for every k, whether each set {n : f(n) = k} has a density, and whether lim sup f(n) is infinite; the paper leaves them open.

Read status. Claims checked for the two pages above, read clause by clause on the print; the proof of (5) was read for its structure only.

Bears on.

• Problem 358: the paper's f(A:n) is the problem's f(n) for a sequence of positive integers. Equation (5) shows that for the primes f(n) = 0 infinitely often, so the primes satisfy neither f(n) -> infinity nor f(n) >= 2 for all large n; Problem 4 asks only whether f(n) is unbounded for the primes. The paper does not mention Erdős or the problem and gives no sequence of the kind it asks for.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.