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Let $K$ be a number field and consider a family of L-functions $\{L(s,\chi_i)\}_{i=1}^N$ associated to Hecke characters of $K$. Assuming the Generalized Riemann Hypothesis (GRH) for this family, we're interested in establishing a "Ratios Conjecture" analogous to the one proposed by Conrey, Farmer, and Zirnbauer for the Riemann zeta function.

Specifically, we aim to find an asymptotic formula for the ratio

$$R_N(\alpha,\gamma) = \frac{1}{N} \sum_{i=1}^N \frac{L(1/2 + \alpha, \chi_i)}{L(1/2 + \gamma, \chi_i)}$$

as $N \to \infty$, for small complex shifts $\alpha$ and $\gamma$.

The main question is whether we can derive a conjectural formula for $R_N(\alpha,\gamma)$ in terms of the arithmetic properties of $K$ and the distribution of the Hecke characters. Of particular interest is how this conjectural formula might relate to the distribution of low-lying zeros of this family of L-functions.

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  • $\begingroup$ What happens for toy cases such as quadratic number fields? $\endgroup$
    – Yemon Choi
    Commented Aug 26 at 3:56

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A ratios conjecture for Hecke characters of $\mathbb Q(i)$ has been stated in Gao & Zhao. Their approach should generalize to other number fields.

The derivation of various properties of low-lying zeros from ratios conjectures is quite general, see e.g. Conrey & Snaith and Mason & Snaith for examples with Dirichlet and elliptic curve $L$-functions. These techniques should generalize to Hecke characters.

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