The ratios conjecture for real Dirichlet characters and multiple Dirichlet series (Q6567170)

In 1993, \textit{D. W. Farmer} [Mathematika 40, No. 1, 71--87 (1993; Zbl 0783.11031)] conjectured \[\N\frac{1}{T}\int_0^T\frac{\zeta(s+\alpha)\zeta(1-s+\beta)}{\zeta(s+\gamma)\zeta(1-s+\delta)}\,dt\sim 1 + (1-T^{-\alpha-\beta}) \frac{(\alpha-\gamma)(\beta-\delta)}{(\alpha+\beta)(\gamma+\delta)},\N\]\Nwhere \(\zeta(s)\) is the Riemann zeta function, \(s=\frac{1}{2}+it\) and the real parts \(\alpha, \beta, \gamma, \delta\) are positive and of the order \(\frac{1}{\log T}\). Based on the conjecture above, \textit{B. Conrey} et al., [Commun. Number Theory Phys. 2, No. 3, 593--636 (2008; Zbl 1178.11056)] came up with the ratios conjectures, which give a general recipe to predict asymptotic formulas for the sum of ratios of products of shifted \(L\)-functions. In this paper under review, the authors studies the family of quadratic Dirichlet \(L\)-functions with one shift in the numerator and in the denominator. For a fundamental discriminant \(d\), let \(\chi_d\) be the Kronecker symbol \(\left(\frac{\cdot}{d}\right)\). In this case, the ratios conjecture is about \N\[\N\sum^{\:\quad*}_{d\leq X}\frac{L(1/2+\alpha,\chi_d)}{L(1/2+\beta,\chi_d)},\N\]\Nwhere the star indicates that the sums run over fundamental discriminants. By assuming the generalized Riemann hypothesis (GRH), the ratio conjecture is established in a smaller range of the shifts.\N\NAs an application, the author computes the one-level density for test functions whose Fourier transform is supported in \((-2, 2)\), including lower-order terms.