gpt pro 5.2 attempt on Math-6027

Does there exist a constant $C>1$ such that, for every $n\geq 2$, there exists a sequence $z_i\in \mathbb{C}$ with $z_1=1$ and $\lvert z_i\rvert \geq 1$ for all $1\leq i\leq n$ with\[\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert < C^{-n}?\] This is Problem 7.3 in [Ha74], where it is attributed to Erdős. Erdős proved (as described on p.35 of [Tu84b]) that such a sequence does exist with $\lvert z_i\rvert\leq 1$. Indeed, Erdős' construction gives a value of $C\approx 1.32$. In [Er92f] (a different) Erdős refines this analysis, proving that if\[M_2=\min_{z_j} \max_{2\leq k\leq n+1} \left\lvert \sum_{1\leq j\leq n}z_j^k\right\rvert,\]where the minimum is take over all $z_j\in \mathbb{C}$ with $\max \lvert z_j\rvert=1$, then\[(1.746)^{-n} < M_2 < (1.745)^{-n}.\]Tang notes in the comments that Theorem 6.1 of [Tu84b] implies that, if $\lvert z_i\rvert \geq 1$ for all $i$, then\[\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert \geq (2e)^{-(1+o(1))n}.\]See also [519] . References [Er92f] Erdős, L., On some problems of {P}. {T}urán concerning power sums of complex numbers . Acta Math. Hungar. $1992$, 11--24. [Ha74] Hayman, W. K., Research problems in function theory: new problems . $1974$, 155--180. [Tu84b] Turán, Paul, On a new method of analysis and its applications . $1984$, xvi+584.

1) FORMAL RESTATEMENT
Question: Does there exist an absolute constant $C>1$ such that for every integer $n\ge 2$ there exist complex numbers $z_1,\dots,z_n$ with \[ z_1=1,\qquad |z_i|\ge 1\ (1\le i\le n), \] and such that \[ \max_{2\le k\le n+1}\Big|\sum_{i=1}^n z_i^k\Big| < C^{-n}? \]

2) QUICK LITERATURE/CONTEXT CHECK
The statement records that Erdős produced sequences under the opposite constraint $|z_i|\le 1$ (and more generally $\max|z_i|=1$) with exponentially small power sums, while Turán's theorem gives an exponential lower bound when $|z_i|\ge 1$. I do not use any other external input.

3) ATTACK PLAN
Proof track: attempt an explicit construction meeting $|z_i|\ge 1$ that forces the first $n$ power sums to be exponentially small; investigate whether one can restrict attention to $|z_i|=1$ without loss, or whether varying radii helps.

Disproof track: attempt to prove a universal lower bound of the form $\max_{2\le k\le n+1}|\sum z_i^k|\ge c^{-n}$ for some $c<1$ that is too large to allow any fixed $C>1$; this seems incompatible with the stated Turán lower bound (which is also exponential), so a disproof would require sharpening.

I only manage a complete analysis for the smallest case $n=2$.

4) WORK
FAST REALITY CHECK (the case $n=2$). For $n=2$ and $z_1=1$, writing $z_2=re^{i\theta}$, I performed a coarse grid search over $r\in[1,4]$ and $\theta$ on a $2000$-point mesh. The minimal value found for \[ \max\big(|1+z_2^2|,\ |1+z_2^3|\big) \] was $0.6180339887\dots$ attained at $r=1$ and $\theta=2\pi/5$. The next proposition proves this is the true optimum.

Lemma 973.1 (monotonicity in the radius for a single term). Fix an integer $k\ge 1$ and an angle $\theta\in\mathbb{R}$. For $r\ge 1$, define \[ F_{k,\theta}(r):=\big|1+r^k e^{ik\theta}\big|. \] Then $F_{k,\theta}(r)$ is nondecreasing for $r\ge 1$.

Proof. Compute the squared modulus: \[ F_{k,\theta}(r)^2 = \big(1+r^k e^{ik\theta}\big)\big(1+r^k e^{-ik\theta}\big)=1+r^{2k}+2r^k\cos(k\theta). \] Differentiate for $r>0$: \[ \frac{d}{dr}F_{k,\theta}(r)^2 = 2k r^{2k-1} + 2k r^{k-1}\cos(k\theta) =2k r^{k-1}\big(r^k+\cos(k\theta)\big). \] For $r\ge 1$, we have $r^k\ge 1$ and $\cos(k\theta)\ge -1$, hence $r^k+\cos(k\theta)\ge 0$, so the derivative is nonnegative. Thus $F_{k,\theta}(r)^2$ is nondecreasing for $r\ge 1$, and therefore so is $F_{k,\theta}(r)=\sqrt{F_{k,\theta}(r)^2}$. $\square$

Proposition 973.2 (exact minimax for $n=2$). Let $n=2$, with $z_1=1$ and $z_2\in\mathbb{C}$ satisfying $|z_2|\ge 1$. Define \[ M_2^{(\ge 1)}:=\inf_{|z_2|\ge 1}\ \max\big(|1+z_2^2|,\ |1+z_2^3|\big). \] Then \[ M_2^{(\ge 1)}=\frac{\sqrt5-1}{2}, \] and the infimum is attained, for instance, at $z_2=e^{2\pi i/5}$ (and by symmetry at $e^{-2\pi i/5}$).

Proof. Write $z_2=re^{i\theta}$ with $r\ge 1$. By Lemma lem:973-radius, for each fixed $\theta$ the quantities $|1+z_2^2|=|1+r^2 e^{2i\theta}|$ and $|1+z_2^3|=|1+r^3 e^{3i\theta}|$ are both nondecreasing in $r$ for $r\ge 1$. Therefore their maximum is minimized at the smallest allowed radius $r=1$. Hence \[ M_2^{(\ge 1)} = \inf_{\theta\in\mathbb{R}} \max\big(|1+e^{2i\theta}|,\ |1+e^{3i\theta}|\big). \] Using $|1+e^{i\varphi}|=2|\cos(\varphi/2)|$, we rewrite \[ |1+e^{2i\theta}|=2|\cos\theta|,\qquad |1+e^{3i\theta}|=2|\cos(3\theta/2)|. \] Let \[ m(\theta):=\max\big(|\cos\theta|,\ |\cos(3\theta/2)|\big),\qquad\text{so }M_2^{(\ge 1)}=2\inf_{\theta} m(\theta). \] The function $m$ is $2\pi$-periodic and even, so it suffices to minimize over $\theta\in[0,\pi]$. At any minimizer $\theta_*$, one must have $|\cos\theta_*|=|\cos(3\theta_*/2)|$; otherwise if (say) $|\cos\theta_*|>|\cos(3\theta_*/2)|$ then by continuity one can perturb $\theta_*$ slightly to decrease $|\cos\theta|$ while keeping $|\cos(3\theta/2)|$ strictly below the previous maximum, reducing $m$, contradicting minimality. Thus we solve the equation \[ |\cos\theta|=|\cos(3\theta/2)|. \] There are two cases.

Case 1: $\cos\theta=\cos(3\theta/2)$. Then either $\theta\equiv 3\theta/2\pmod{2\pi}$ (giving $\theta\equiv 0$) or $\theta\equiv -3\theta/2\pmod{2\pi}$ (giving $5\theta/2\equiv 0\pmod{2\pi}$, hence $\theta\in\{0,4\pi/5\}$ within $[0,\pi]$). At $\theta=0$, $m(0)=1$. At $\theta=4\pi/5$, $|\cos\theta|=|\cos(4\pi/5)|=|\cos(\pi-\pi/5)|=\cos(\pi/5)=\frac{\sqrt5+1}{4}\approx 0.809$. So this case yields $2m\ge 2\cdot \frac{\sqrt5+1}{4}=\frac{\sqrt5+1}{2}\approx 1.618$.

Case 2: $\cos\theta=-\cos(3\theta/2)$. Using $\cos a+\cos b=2\cos\frac{a+b}{2}\cos\frac{a-b}{2}$, the equation becomes \[ \cos\theta+\cos(3\theta/2)=0 \iff 2\cos\Big(\frac{5\theta}{4}\Big)\cos\Big(\frac{\theta}{4}\Big)=0. \] Within $\theta\in[0,\pi]$, the factor $\cos(\theta/4)$ is positive, so we must have $\cos(5\theta/4)=0$, i.e. \[ \frac{5\theta}{4}=\frac{\pi}{2}+\pi m \quad\Longrightarrow\quad \theta=\frac{2\pi}{5}(1+2m). \] The unique solution in $[0,\pi]$ is $\theta=2\pi/5$. At this angle, \[ |\cos\theta|=\cos(2\pi/5)=\cos(72^\circ)=\frac{\sqrt5-1}{4}. \] Hence $m(2\pi/5)=\frac{\sqrt5-1}{4}$ and the corresponding objective value is \[ 2m(2\pi/5)=\frac{\sqrt5-1}{2}. \] Comparing with Case 1 (and boundary values), this is the global minimum. Thus $M_2^{(\ge 1)}=(\sqrt5-1)/2$, achieved at $z_2=e^{i\theta}=e^{2\pi i/5}$. $\square$

5) VERIFICATION
Lemma lem:973-radius was checked by explicit differentiation; the key inequality $r^k+\cos(k\theta)\ge r^k-1\ge 0$ holds for $r\ge 1$. In Proposition prop:973-n2, the reduction to $r=1$ is valid because both competing terms are separately nondecreasing in $r$ for fixed $\theta$. The trigonometric minimization checks all solutions to the equal-max condition and compares values.

6) FINAL
UNRESOLVED

(i) Strongest proved partial result. For $n=2$ the exact minimax value under the constraint $|z_2|\ge 1$ is $M_2^{(\ge 1)}=(\sqrt5-1)/2$ (Proposition prop:973-n2), achieved on the unit circle. Also, for each individual term $|1+r^k e^{ik\theta}|$ is minimized at the smallest allowed radius $r=1$ (Lemma lem:973-radius).

(ii) First gap (crisp). For general $n$, construct (or rule out) sequences $z_1=1$, $|z_i|\ge 1$ such that \[ \max_{2\le k\le n+1}\Big|\sum_{i=1}^n z_i^k\Big|\le C^{-n} \] for some absolute constant $C>1$ independent of $n$.

(iii) Top 3 next moves.

• Determine whether one may assume $|z_i|=1$ without loss for the minimax problem under $|z_i|\ge 1$; Lemma lem:973-radius shows this for $n=2$ but not for coupled sums.
• Attempt an explicit construction: choose $z_i$ as (possibly repeated) points on or outside the unit circle designed to make the first $n$ power sums small simultaneously (e.g. via polynomial/Newton-identity control of power sums), while keeping $|z_i|\ge 1$.
• Compute numerical optima for small $n$ with constrained optimization (e.g. restricting to $|z_i|=1$ and varying phases) to guess the correct exponential base for the $|z_i|\ge 1$ constraint.

(iv) Minimal counterexample structure. A negative answer would mean: for every $C>1$ there are infinitely many $n$ such that for all choices with $z_1=1$ and $|z_i|\ge 1$, \[ \max_{2\le k\le n+1}\Big|\sum_{i=1}^n z_i^k\Big|\ge C^{-n}. \] In other words the minimax quantity for the $|z_i|\ge 1$ constraint would decay at best subexponentially (or with base $\le 1$), contradicting the heuristic suggested by the stated Turán-type exponential lower bound; any such counterexample would likely hinge on a coupled constraint preventing simultaneous cancellation across the $n$ different exponents $k=2,\dots,n+1$.