On a theorem of Levinson
Let \(F(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)\), where \(\zeta(s)\) is the Riemann zeta-function. \textit{N. Levinson} [J. Number Theory 3, 502--504 (1971; Zbl 0224.10042)] proved a lower bound for the number of zeros of \(\operatorname{Re} F(1/2+\lambda+it)\) with \(0<t\leq T\) and fixed \(\lambda>0\); his approach relied on the classical density theorem \(N(1/2+\lambda,T)\ll T^{1-\lambda/4}\log T\), where the left hand side counts the number of zeros of \(\zeta(s)\) in the region \(\operatorname{Re} s>1-\lambda/4,0<t\leq T\). Define \[ F_\lambda^\pm(z)={\Xi(z-i\lambda)\over (z-i\lambda)^2+{1\over 4}}\pm {\Xi(z+i\lambda)\over (z+i\lambda)^2+{1\over 4}},\qquad\text{where}\quad \Xi(s):={s(s-1)\over 2}F(s). \] Then \(F_\lambda^+(t)=-\operatorname{Re} F(1/2+\lambda+it)\) and \(F_\lambda^-(t)=-i\operatorname{Im} F(1/2+\lambda+it)\). In the present paper, the author proves the following complementary statement to Levinson's estimate: Let \(\lambda>0\). There exists a constant \(\kappa>0\) such that all zeros of \(F_\lambda^\pm(z)\) which are in \[ \{z: |\operatorname{Re} z| \geq|\operatorname{Im} s_1| \;\text{and}\;|\operatorname{Re}z -\text{Im}s_k^*|>\kappa,s_k^*=s_k\;\text{or}\;\overline{s_k}\;\text{for}\;k=1,2,\ldots\} \] are real; in particular, if \(\lambda\geq 1/2\), then all zeros of \(F_\lambda^\pm(z)\) are real. Here \(s_1, s_2,\ldots\) denotes the zeros of \(\zeta(s)\) with real part greater than \(1/2+\lambda\) and \(0<\operatorname{Im} s_1\leq \operatorname{Im} s_2\leq \ldots\), in case they exist. As a corollary, the author obtains that, for \(\lambda\in(0,1/2)\), the number of real zeros of \(F_\lambda(z)\) within \(0<\operatorname{Re} z\leq T\) is asymptotically equal to \(T/(2\pi)\log(T/(2\pi e))+O(N(1/2+\lambda,T)\log T)\). Furthermore, the author claims that the method of proof shows that all zeros are real for any \(\lambda>0\) if the Riemann hypothesis is true.
scientific article; zbMATH DE number 2091712