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Statement

Lemma 1 (pp. 8–9). Let $M$ be a positive integer and $\theta$ a real number with $\theta\not\equiv0\pmod{2\pi}$. Then

$$\log\bigl|1-e^{i\theta}\bigr|\le-\sum_{m=1}^{M-1}\Bigl(1-\frac mM\Bigr)^2 \frac{\cos m\theta}{m}+M^{-2}(2M-1)\log2.\qquad(10)$$

For $M=1$ the sum is empty, the bound reads $\log|1-e^{i\theta}|\le\log2$, and the paper notes that it is attained at $\theta=\pi$ (p. 9).

The lemma replaces the Fourier series $\log|1-e^{i\theta}|=-\sum_{m\ge1}m^{-1}\cos m\theta$ (p. 8, (7)), which does not converge absolutely, by a partial sum with weights $(1-m/M)^2$ at the cost of the additive error $M^{-2}(2M-1)\log2$.

Source. Lemma 1, stated on p. 8 with display (10) on p. 9 and proved on pp. 9–10, of F. V. Atkinson, On a problem of Erdős and Szekeres, Canad. Math. Bull. 4 (1961), 7–12, DOI 10.4153/CMB-1961-002-5, as identified on the source card.

Read depth. Claims checked: the statement and its hypotheses were read clause by clause on pp. 8–9; the proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

pp. 9–10. It suffices to take $0<\theta\le\pi$. Integrating $1/(1-z)$ from $e^{i\theta}$ to $0$, the paper writes $\log|1-e^{i\theta}|$ as the partial sum of order $N$ plus an integral round the unit circle and one along $[-1,0]$ (its (11)), then averages (11) over $N=0,\ldots,M-1$ with weights $2M-1-2N$ (its (12)). The circle term is non-negative because its weighted cosine sum equals $\cos\frac12\theta\,\sin^2\frac12M\theta\,\operatorname{cosec}^2\frac12\theta$, and the real-axis term is at most $(2M-1)\log2$ because for $-1<z<0$ the terms of $\sum_{N=0}^{M-1}(2M-1-2N)z^N$ alternate in sign and decrease in absolute value, so the sum lies between $0$ and $2M-1$.

Bears on

• Problem 256: through Lemma 2, the lemma is the analytic input of inequality (5); on its own it states no bound for $f(n)$.