Autocorrelation of ratios of \(L\)-functions
Methods from random matrix theory have proved to be very useful in providing conjectures for the main terms of many expressions involving the averages (mean values) of various \(L\)-functions [see e.g., \textit{J. B. Conrey} et al., Proc. Lond. Math. Soc. (3) 91, No. 1, 33--104 (2005; Zbl 1075.11058)]. In the present work the authors give a new heuristic for all of the main terms in the quotient of products of \(L\)-functions averaged over a family. As a model one may mention the \textit{First Ratios Conjecture}, made by the second author [Mathematika 40, No. 1, 71--87 (1993; Zbl 0783.11031)] \[ {1\over T}\int_0^T{\zeta(s+\alpha)\zeta(1-s+\beta)\over \zeta(s+\gamma)\zeta(1-s+\delta)}\,dt\sim 1 + (1-T^{-\alpha-\beta}) {(\alpha-\gamma)(\beta-\delta)\over(\alpha+\beta)(\gamma+\delta)},\leqno(1) \] where the real parts of \(\alpha, \beta, \gamma, \delta\) are positive and of the order \(1/\log T\). In Section 5.2 the authors assume the Riemann Hypothesis and conjecture a formula that generalizes (1), involving a ratio of products of several zeta-functions, and they even state that the error term should be \(O(T^{1/2+\varepsilon})\). In Section 3 they survey the \(L\)-functions that are considered later, namely the Riemann zeta-function \(\zeta(s)\), Dirichlet \(L\)-functions with real characters, quadratic twists of modular \(L\)-functions. Autocorrelations of ratios of characteristic polynomials, which serve as a model after which the authors' conjectures are made, are investigated in Section 4, where three theorems are proved (unconditionally). Refinements of the conjectures as well as several applications are also given. In particular, a more precise version of Farmer's conjecture (1) is stated in Section 7, as well a conjecture involving the mean square of the logarithmic derivative of \(\zeta(s)\). The ``recipe'' for making conjectures in Section 5.1 is clearly stated. Therefore this paper will prove useful in making conjectures for various other averages of ratios of \(L\)-functions.
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