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Statement

Inequality (3) (p. 185). Let $z_1,\ldots,z_n$ be complex numbers with

$$1=z_1\ge\lvert z_2\rvert\ge\cdots\ge\lvert z_n\rvert, \tag{1}$$

and put $s_k=\sum_{m=1}^n z_m^k$ and $s=\max_{1\le k\le n}\lvert s_k\rvert$, the paper's (2). Then

$$s>1/6. \tag{3}$$

The bound holds for every $n$ and every choice of $z_1,\ldots,z_n$ subject to (1). The paper presents it, in its words (p. 185), "Without any suggestion that this is a precise value"; it makes no claim that $1/6$ is best possible.

Context (p. 185). Turán posed the problem of a positive lower bound for $s$ valid for all choices subject to (1). The paper records Turán's bound $(\log 2)/\sum_{m=1}^n m^{-1}$, de Bruijn's improvement to $C\log\log n/\log n$ for some $C>0$ and sufficiently large $n$, and Uchiyama's proof that $C$ could be taken arbitrarily close to $1$; (3) verifies the conjecture that $s$ has a positive lower bound independent of $n$.

Source. F. V. Atkinson, On sums of powers of complex numbers, Acta Math. Acad. Sci. Hungar. 12 (1961), no. 1--2, 185--188, DOI 10.1007/BF02066680: (1), (2) and (3) on p. 185, the proof on pp. 185--188, (13) on p. 187 and the concluding deduction on p. 188. The edition read is identified on the source card.

Read depth. Claims checked: (1), (2), (3) and the inequality (13) with its hypothesis $s<1/4$ were read clause by clause on the page images. The proof was read for its structure, not checked line by line. Nothing here is independently reviewed.

Proof pointer

Sections 2 and 3, pp. 185--188. With $g(\theta)=-\sum_{m=1}^n m^{-1}s_m e^{mi\theta}$, the exponential $e^{g(\theta)}$ equals $\prod_{r=1}^n(1-z_re^{i\theta})$ plus a power series in $e^{i\theta}$ starting at the exponent $n+1$ (equations (4), (5)); since $z_1=1$, the product vanishes at $\theta=0$. Reading the tail coefficients as Fourier coefficients and integrating by parts gives the identity (8),

$$1=(2\pi i)^{-1}\int_{-\pi}^{\pi}g'(\theta)e^{g(\theta)-g(0)}h(\theta)\,d\theta, \qquad h(\theta)=\sum_{m=n+1}^{\infty}m^{-1}e^{-mi\theta}.$$

Schwarz's inequality, Parseval's equality for $g'$ (giving at most $2\pi ns^2$), and separate bounds for the second factor on $\lvert\theta\rvert\le\pi/n$ and $\pi/n\le\lvert\theta\rvert\le\pi$, the latter assuming $s<1/4$, give (13) (p. 187):

$$1<s^2e^{2\pi s}\{1+e^{4s}(1-4s)^{-1}\}. \tag{13}$$

The right side is independent of $n$ and increasing on $0<s<1/4$, and (13) fails at $s=1/6$ (p. 188), so $s>1/6$. Not reconstructed further here.

Dependencies

None beyond classical analysis: the expansion (4), which the paper takes from Uchiyama (Acta Math. Acad. Sci. Hungar. 9 (1958), 275--278), Schwarz's inequality and Parseval's equality.

Bears on

• Problem 519: the problem asks whether $\max_{1\le k\le n}\lvert\sum_i z_i^k\rvert$ exceeds an absolute constant $c>0$ for all complex $z_1,\ldots,z_n$ with $z_1=1$. Inequality (3) gives $c=1/6$ under the paper's condition (1), which adds that every $\lvert z_m\rvert$ is at most $1$. The problem's hypothesis reduces to (1) (an observation of this page, not of the paper): dividing every $z_m$ by one of largest modulus $M\ge1$ divides $\lvert s_k\rvert$ by $M^k$, and after relabeling the quotients satisfy (1), so the bound $1/6$ holds under the problem's hypothesis too. The problem's claim page for this paper records the result.