Turán's work in analytic number theory (Q2874927)

A survey is presented of the work of Paul Turán in analytical number theory. The author describes the relation between zero-free regions of the Riemann zeta-function and the size of the error term in the prime number theorem proved by \textit{P. Turán} [Acta Math. Acad. Sci. Hung. 1, 155--166 (1950; Zbl 0041.37102)], and then mentions two results of \textit{S. Knapowski} and \textit{P. Turán} [Acta Math. Acad. Sci. Hung. 14, 241--250, 251--268 (1963; Zbl 0117.27805); Acta Math. Acad. Sci. Hung. 14, 31--42, 43--63, 65--78 (1963; Zbl 0117.03105)] in the comparative prime number theory created by these authors, which compares the behaviour of primes in different arithmetical progressions. The next subject forms the zero density theorem, bounding \(N(\sigma,T)\), the number of zeros \(\beta+i\gamma\) of \(\zeta(s)\) in rectangles \(\beta\geq\sigma, 0<\gamma\leq T\). It is conjectured (the density hypothesis) that one has \(N(\sigma,T)=O(T^{2(1-\sigma)}\log^AT)\) (with some \(A\)), and \textit{P. Turán} [Acta Math. Acad. Sci. Hung. 2, 39--73 (1951; Zbl 0044.03802)] established the bound \(N(\sigma,T)=O(T^{2(1-\sigma)+600(1-\sigma)^c})\) with \(c=1.01\). The last two topics recall Turán's results dealing with the zeros of partial sums of the series for \(\zeta(s)\), Goldbach conjecture and the Erdős-Turán inequality for trigonometric polynomials.NEWLINENEWLINEFor the entire collection see [Zbl 1279.00053].