An improvement of the Erdős-Turán theorem on the distribution of zeros of polynomials (Q2476525)

Let \(f(z)=a_nz^n+\cdots+a_0\) be a complex polynomial of degree \(n\), not vanishing at \(0\). In 1950, \textit{P. Erdős} and \textit{P. Turán} [Ann. Math. (2) 51, 105--119 (1950; Zbl 0036.01501)] showed that the number \(N_f(\alpha)\) of zeros of \(f\) lying in a sector \(\Omega(\alpha)\) of angle \(\alpha\) rooted at \(0\) differs from \(n\alpha/2\pi\) by at most \(c\sqrt n\), where \[ c={16\over\sqrt{| a_0a_n| }}\max_{| z| =1}| f(z)| . \] The author obtains an one-sided analogue of this result for the number \(N_-(\alpha)\) and \(N_+\) of zeros \(z\) of \(f\) lying in \(\Omega(\alpha)\) with \(| z| \leq1\) and \(| z| \geq1\), respectively.