Some observations on the Saff-Varga width conjecture (Q1178599)

Let \(0\leq\tau<\infty\), \(K>0\), and \(x_0\geq 0\). Define \(S(\tau,K;x_ 0)=[z=x+iy\): \(| y|\leq Kx^{1-\tau/2}\), \(x\geq x_0]\). If \(f\) is an entire function of order \(\lambda\) where \(\tau<\lambda\leq\infty\), then the Saff-Varga width conjecture [cf. \textit{E. B. Saff} and \textit{R. S. Varga}, Pac. J. Math. 62, 523--549 (1976; Zbl 0335.30028)] is that the region \(S(\tau,K;x_0)\) contains infinitely many zeros of the partial sums of the Taylor series for \(f\) about the origin. The author notes that for an entire function \(f\) of order \(\lambda\) with \(0<\lambda\leq\infty\) defined by \(f(z)=\sum_{k=0}^\infty a_k z^k\) the conjecture may be restated in terms of the following sufficient conditions: there exists a sequence \(\{z_{n_k}\}\) of zeros of the partial sums \(\{s_{n_k}\}\) of \(f\) satisfying \[ \lim| z_{n_k}|=\infty,\quad (k\to\infty); \tag{I} \] \[ \lim(\log| z_{n_k}|)/(\log n_ k)\leq 1/\lambda,\quad (k\to\infty); \tag{II} \] and \[ \limsup(\log|\arg z_{n_ k}|)/(\log n_ k)\leq -1/2,\quad (k\to\infty). \tag{II} \] While sequences satisfying conditions (I) and (II) are known, the remaining condition (III) has eluded the author's gasp, though theorems are proved which show that a possible route to this construction might be through the proof of a modification of an earlier result of \textit{P. Erdős} and \textit{P. Turán} [Ann. Math. (2) 51, 105--119 (1950; Zbl 0036.01501)].