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Statement

Setting (p. 62). As in Theorem 1: $f(z)=\sum_{n\ge0}a_nz^{\lambda_n}$ is entire, (1), with $\lambda_n$ a strictly increasing sequence of non-negative integers, and $M(r)$, $m(r)$ and $\mu(r)$ are its maximum modulus, minimum modulus and maximum term.

Theorem 4 (p. 63). Suppose that, as $n\to\infty$, either

$$\sum_{k=0}^n\frac{1}{\lambda_{k+1}-\lambda_k}=o(\log\lambda_n)\qquad(12)$$

and $f$ has finite order, or

$$\sum_{k=0}^n\frac{1}{\lambda_{k+1}-\lambda_k}=O(\log\lambda_n)\qquad(13)$$

and $f$ has zero order. The print's conclusion reads "then (2) holds" [sic]. Display (2) is Pólya's gap condition, a hypothesis on $\lambda_n$ alone, so the reference cannot be meant literally. The proof (pp. 68--70) ends by showing that a single term of $f$ dominates the rest of the series on circles $\lvert z\rvert=RA_N$ with $N$ arbitrarily large, which is the property from which the proof of Theorem 1 derives (3), $\limsup m(r)/M(r)=\limsup\mu(r)/M(r)=1$; the paper introduces the theorem as relaxing the gap condition of Theorem 1 at the cost of an order condition (p. 63). The reading of the conclusion as (3) is this page's, not the print's.

The paper says (pp. 63--64) that the theorem cannot be materially strengthened, citing the order of the function constructed for Theorem 2.

Proof pointer

Pp. 68--70, Section 5. For a small $\delta>0$ the paper builds an auxiliary series $\sum c_nx^{\lambda_n}$ with positive coefficients and radii $A_N$ at which consecutive terms are in ratio $\delta$, (46)--(52), so one term dominates, (49)--(50). Since $\log A_n$ is $\log K$ times a partial reciprocal gap sum, (53), the auxiliary series is entire when $A_n\to\infty$, which requires the full sum to diverge. Comparing $f$ with it, the domination carries over to $f$ when $\sum a_nz^{\lambda_n}/c_n$ is entire, (54); for finite order this follows from (12) through (55)--(58), and for zero order from (13).

Read depth

Claims checked: Theorem 4, (12), (13) and the remark on sharpness were read clause by clause on the page images of the print, and the proof on pp. 68--70 was followed for structure. The conclusion is a misprint in the print, read here as described above. Nothing here is independently reviewed.

Dependencies

The conclusion is that of Theorem 1, and the sharpness remark uses the construction of Theorem 2.

Source. P. Erdős and A. J. Macintyre, Integral functions with gap power series, Proc. Edinburgh Math. Soc. (2) 10 (1954), 62--70; the edition read is named on the source card.

Bears on

• Problem 516: the theorem is the paper's criterion for functions of finite order, the problem's class, under the gap condition (12), which the paper presents as a relaxation of the convergent sum (4) of Theorem 1. Read with the conclusion (3), it gives $\limsup m(r)/M(r)=1$ on the functions it covers, stronger than the problem's $\limsup\log m(r)/\log M(r)=1$. The paper does not treat every finite-order function with $\lambda_n/n\to\infty$.