Critical zeros of Dirichlet \(L\)-functions (Q2848380)

Let \(\chi\) \(\pmod{q}\) be a primitive character, and \(L(s,\chi)\) the associated Dirichlet series. Further, let \(N(T,\chi)\) denote the number of complex zeros \(\rho = \beta+i\gamma\) of \(L(s,\chi)\) for which \(0<\beta< 1, |\gamma| \leq T\), \(N_0(T,\chi)\) the number of zeros counted by \(N(T,\chi)\) for which \(\beta = 1/2\) (these are the zeros on the so-called ``critical line''), and \(N_0^{'}(T,\chi)\) the number of simple zeros counted by \(N_0(T,\chi)\). In this paper, the authors prove, among other things, the remarkable result that at least 56\ for \(q = 1\) includes the case of the classical Riemann zeta-function \(\zeta(s)\). This is a substantial improvement over the previous records, namely 41.05\ [\textit{H. M. Bui} et al., Acta Arith. 150, No. 1, 35--64 (2011; Zbl 1250.11083)] and \textit{S. Feng} [J. Number Theory 132, 511--542 (2012; Zbl 1333.11086)], who had 41.27.NEWLINENEWLINELet \(\Psi(x)\) be a non-negative, smooth, compactly supported function on \({\mathbb R}^+\). Put NEWLINE\[CARRIAGE_RETURNNEWLINE {\mathcal N}(T,Q) = \sum_q\frac{\Psi(q/Q)}{\phi(q)}\sum_{\chi}\,\pmod{q}^{*}N(T,\chi) CARRIAGE_RETURNNEWLINE\]NEWLINE for \(Q\geq3\) and \( T\geq3\), where \({}^*\) denotes summation over primitive characters. Let \(\mathcal {N}_0^{'}(T,Q)\) denote the same expression, but with \(N(T,\chi)\) replaced by \(N_0^{'}(T,\chi)\).NEWLINENEWLINEThe authors' result, Theorem 1, says that, for \(Q\) and \(T\) satisfying \((\log Q)^6 \leq T\leq (\log Q)^A, A\geq 6\) a constant, NEWLINE\[CARRIAGE_RETURNNEWLINE \mathcal {N}_0^{'}(T,Q) \,\geq\, \frac{14}{25}{\mathcal N}(T,Q)\qquad(T\geq T_0 >0, Q \geq Q_0 >0). CARRIAGE_RETURNNEWLINE\]NEWLINENEWLINENEWLINEIn the Appendix, the authors expound the basics of the famous method of \textit{N. Levinson} [Adv. Math. 13, 383--436 (1974; Zbl 0281.10017)], who showed that more than 1/3 of the zeros of \(\zeta(s)\) lie on the critical line. With some modifications this is the method used in the present paper.NEWLINENEWLINEThe authors explain how in the course of their proofs one needs the evaluation of integrals of the type NEWLINE\[CARRIAGE_RETURNNEWLINE I_\chi := \int|G(\sigma +it,\chi)M(\tfrac 12 +it,\chi)|^2\,\Phi(t)\,dt. CARRIAGE_RETURNNEWLINE\]NEWLINE The smooth function \(\Phi(t)\) satisfies certain conditions, and NEWLINE\[CARRIAGE_RETURNNEWLINE G(s,\chi) := L(s,\chi) + \lambda L'(s,\chi) CARRIAGE_RETURNNEWLINE\]NEWLINE with a suitable constant \(\lambda\). The ``mollifier'' \(M(s,\chi)\) is NEWLINE\[CARRIAGE_RETURNNEWLINE M(s,\chi) := \sum_{n\leq X}\mu(n)\chi(m)m^{-s}P\left(1 - \frac{\log m}{\log X}\right), CARRIAGE_RETURNNEWLINE\]NEWLINE and \(\mu(\cdot)\) is the familiar Möbius function. Here \(P(x)\) is a smooth function with \(P(0) =0\) and \(P(1)\).NEWLINENEWLINETheorem 2 is lengthy and technical. It gives an asymptotic formula for \(I_\chi\), averaged over \(\chi\) and \(q\).NEWLINENEWLINEFinally, Theorem 3 represents an analogue of Theorem 1 for \(\mathrm{GL}_2\) \(L\)-functions and \(\mathrm{GL}_3\) \(L\)-functions. It says that, on the average, 35\ \(\mathrm{GL}_2\) \(L\)-function are simple and lie on the critical line, and 0.5\ a given \(\mathrm{GL}_3\) \(L\)-function are simple and lie on the critical line.