Zeros of \(L(s,\chi)\) in short segments on the critical line (Q1109812)

Let \(L(s,\chi)\) be a Dirichlet L-function with \(\chi\) primitive (mod k), \(k>1\), and let \(N_ 0(T,\chi)\) denote the number of zeros \(\rho\) of \(L(s,\chi)\) of the form \(\rho =+i\gamma\), \(0<\gamma \leq T\). The author gives a sketch of proof of the following important result: Let \(T\geq k^{+6\epsilon}\), \(U\geq (kT)^{1/3+2\epsilon}\) with small \(\epsilon >0\). Then we have \[ N_ 0(T+U,\chi)-N_ 0(T,\chi)\quad \gg_{\epsilon}\quad U \log T. \] This result should be compared to \textit{A. A. Karatsuba}'s result [Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 3, 569-584 (1984; Zbl 0545.10026)] for \(\zeta(s)\). The idea of proof is based on \textit{A. Selberg}'s classical approach [Skr. Nor. Vidensk.-Akad., I. Mat.-Naturv. Kl., N. Ser., No. 10, 1-59 (1943; Zbl 0028.11101)] of dealing with the zeros of \(\zeta\) (s) on the critical line. But the author skilfully uses some of his own ideas, and the method of \textit{F. V. Atkinson} [Acta Math. 81, 353-376 (1949; Zbl 0036.186)] for the study of the mean square of \(| \zeta (+it)|.\) The present work is related to the author's five notes on the mean values of the zeta- and L-functions [Proc. Japan Acad., Ser. A 61, 222-224 (1985; Zbl 0573.10027), ibid. 61, 313-316 (1985; Zbl 0583.10024), ibid. 62, 152-154 (1986; Zbl 0592.10037), ibid. 62, 311-313 and 399-401 (1986; Zbl 0613.10032)].