Erdos 1954 integral functions gap power series

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theorem_1: Erdős and Macintyre's theorem that an entire function sum a_n z^{lambda_n} whose reciprocal gaps 1/(lambda_{n+1} - lambda_n) have a convergent sum satisfies limsup m(r)/M(r) = limsup mu(r)/M(r) = 1, with no order hypothesis.

theorem_2: Erdős and Macintyre's theorem that when the reciprocal gaps 1/(lambda_{n+1} - lambda_n) have a divergent sum there is an entire function sum a_n z^{lambda_n} with limsup mu(r)/M(r) <= 1/2 and limsup m(r)/M(r) <= 1/2, so their Theorem 1 is best possible.

theorem_3: Erdős and Macintyre's theorem that convergence of sum 1/(lambda_{n+h} - lambda_n) for a positive integer h gives limsup mu(r)/M(r) >= 1/(2h-1), while divergence for every h permits an entire function with lim mu(r)/M(r) = lim m(r)/M(r) = 0.

theorem_4: Erdős and Macintyre's theorem for an entire function sum a_n z^{lambda_n} of finite order whose partial reciprocal gap sums are o(log lambda_n), or of zero order with those sums O(log lambda_n); the print states its conclusion as (2), Pólya's gap condition, and its proof gives the single-term dominance behind the conclusion (3) of Theorem 1.

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P. Erdős, A. J. Macintyre: Integral functions with gap power series, Proc. Edinburgh Math. Soc. (2) 10 (1954), 62--70 (MR 16,579a; Zentralblatt 58,63).

For an entire function f(z) = sum a_n z^{lambda_n} the authors study when limsup m(r)/M(r) = 1, where M and m are the maximum and minimum modulus and mu the maximum term. Theorem 1 shows that the convergence of sum 1/(lambda_{n+1} - lambda_n) suffices, sharpening Pólya's condition liminf log(lambda_{n+1} - lambda_n)/log lambda_n > 1/2, and Theorem 2 constructs a counterexample when that sum diverges, so Theorem 1 is best possible. Theorem 3 replaces consecutive gaps by h-step gaps: convergence of sum 1/(lambda_{n+h} - lambda_n) forces limsup mu(r)/M(r) >= 1/(2h-1), while divergence for every h permits an entire function with lim mu(r)/M(r) = lim m(r)/M(r) = 0; the stronger guess limsup m(r)/M(r) > 0 under the h-gap hypothesis is disproved by an explicit two-block example. Theorem 4 instead adds an order hypothesis: it covers f of finite order with sum_{k<=n} 1/(lambda_{k+1}-lambda_k) = o(log lambda_n), and f of zero order with the same sum O(log lambda_n). Its printed conclusion, "then (2) holds" [sic], names Pólya's gap condition; its proof gives the single-term dominance from which Theorem 1's proof derives (3). The paper says Theorem 4 cannot be materially strengthened, by the order of the Theorem 2 example (pp. 63--64). Problem 516 asks whether lambda_n/n -> infinity gives limsup log m(r)/log M(r) = 1 for entire functions of finite order; Theorem 1 gives the stronger limsup m(r)/M(r) = 1 under its gap condition and any order, and Theorem 4 gives it, read with (3), for finite order under its weaker gap condition.

Source: https://users.renyi.hu/~p_erdos/1954-01.pdf. No notice is printed on pp. 62--63 or 69--70, and the card records no DOI, so the article's own page was not resolved; the publisher's journal page names Cambridge University Press, prints the footer "Cambridge University Press 2026", marks the journal "Contains open access" and names no license for back volumes (https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society, read 2026-10-02), every other right reserved.

Bears on. #516: Theorem 1 (p. 62) gives limsup m(r)/M(r) = 1, stronger than the problem's limsup log m(r)/log M(r) = 1, for every entire function, of any order, whose reciprocal gaps have a convergent sum; this covers only part of the problem's class, as the problem's claim page for this paper records. Theorem 4 (p. 63) is the paper's finite-order criterion under the weaker gap condition (12), with its conclusion printed as "then (2) holds" [sic]. Theorem 2 (pp. 62--63) shows the conclusion limsup m(r)/M(r) = 1 can fail when the gap sum diverges; it says nothing about the logarithmic ratio. The paper does not answer the problem for every finite-order function with lambda_n/n -> infinity.

Results.

• Theorem 1 (p. 62): if sum_n 1/(lambda_{n+1} - lambda_n) converges then limsup m(r)/M(r) = limsup mu(r)/M(r) = 1, sharpening Pólya's gap condition (2).
• Theorem 2 (pp. 62--63): if that gap sum diverges there is an entire function of the form (1) with limsup mu(r)/M(r) <= 1/2 and limsup m(r)/M(r) <= 1/2, so Theorem 1 is best possible.
• Theorem 3 (p. 63): convergence of sum_n 1/(lambda_{n+h} - lambda_n) for a positive integer h gives limsup mu(r)/M(r) >= 1/(2h-1); if the sum diverges for every h there is an entire function of the form (1) with lim mu(r)/M(r) = lim m(r)/M(r) = 0.
• Theorem 4 (p. 63): if sum_{k<=n} 1/(lambda_{k+1} - lambda_k) = o(log lambda_n) and f has finite order, or the same sum is O(log lambda_n) and f has zero order, then "(2) holds" [sic]; the proof gives the single-term dominance behind (3).

Read status: claims checked. Theorems 1 to 4, (2), (3) and the remarks on pp. 63--64 were read clause by clause on the page images of the print, and the proofs were followed for structure; the paper leaves the last step of the second part of Theorem 3 to the reader. Nothing here is independently reviewed.

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