Ratios Conjecture CFZ: Zeta, L-functions & RMT

• The paper provides a comprehensive recipe for predicting asymptotic averages of ratios of L-functions using combinatorial sums and explicit arithmetic corrections.
• It employs differentiation of the ratios to extract n-correlation functions, integrating prime-dependent local factors for detailed zero statistics.
• The framework unifies the treatment of zeta and L-functions with random matrix theory by accurately capturing both leading and lower-order (arithmetic) terms.

The Ratios Conjecture of Conrey, Farmer, and Zirnbauer (CFZ) is a conjectural framework designed to predict the precise asymptotic behavior of averages of ratios of products of shifted zeta- or L-functions across families. The conjecture has played a pivotal role in sharpening the connection between analytic number theory and random matrix theory (RMT), allowing for the explicit calculation not only of leading-order terms in statistical properties of zeta zeros and L-function zeros, but also of all lower-order (arithmetic) corrections. This approach enables the direct and systematic extraction of n-correlation functions for Riemann zeros and eigenvalue statistics for classical compact groups.

1. The Ratios Conjecture: Formulation and Predictive Methodology

At its core, the Ratios Conjecture provides a detailed “recipe” for predicting the mean value of ratios such as

$$R(A,B;C,D) = \left\langle \frac{\prod_{a\in A} \zeta(s+a)\ \prod_{c\in C} \zeta(1-s+c)} {\prod_{b\in B} \zeta(s+b)\ \prod_{d\in D} \zeta(1-s+d)} \right\rangle$$

where $A,B,C,D$ are (multi-)sets of finite shifts. This average is taken over an appropriate region of the critical line (or, for RMT, over the Haar measure on $U(N)$).

The conjecture expresses $R(A,B;C,D)$ as a combinatorial sum over subsets $S\subseteq A$ and $T\subseteq B$ with $|S|=|T|$, where each term consists of an exponential “global” factor, an arithmetic product $Z(S,T;C,D)$ encoding local data through zeta- and prime-dependent contributions, and—after differentiation with respect to the shifts—factors that encapsulate dependencies among shifts.

Explicitly, the schematic formula is: $$R(A,B;C,D) = \sum_{\substack{S\subset A\ T\subset B \ |S|=|T|}} \exp\!\left(-N\Bigl(\sum_{\hat a\in S} \hat a + \sum_{\hat b\in T} \hat b\Bigr)\right) \; Z(S, T; C, D)$$ where $N$ plays the role of a (scaling) parameter (e.g., $N = \log T/(2\pi)$ for zeta zeros, or $N$ for the size of a random matrix).

2. Differentiation and Correlations of Logarithmic Derivatives

Upon differentiating $R(A,B;C,D)$ with respect to the shifts, one obtains averages of products of logarithmic derivatives of the zeta function: $$J(A;B) = \left\langle \prod_{a\in A} \frac{\zeta'}{\zeta}(s+a) \prod_{b\in B} \frac{\zeta'}{\zeta}(s+b) \right\rangle$$ This average, central to the extraction of zero statistics, is expressed as

$$J^*(A;B) = \sum_{\substack{S\subset A\ T\subset B \ |S|=|T|}} \exp\left(-N\left(\sum_{\hat a \in S} \hat a + \sum_{\hat b\in T} \hat b\right)\right) Z(S,T)\ Z(S^-,T^-)\ \mathcal{H}_{S,T}$$

where $Z(\cdot)$ are products of shifted zeta factors (with local corrections as in Conjecture 1), $S^-$ denotes the set of negatives of the shifts in $S$, and $\mathcal{H}_{S,T}$ encodes the contribution from the local differentiation structure. Primes enter through local factors such as $z_p(x) = (1-p^{-1-x})^{-1}$ as well as more elaborate arithmetic objects $Z_p$ and $A_{p,0}$.

3. Explicit n-Correlation Formulas and Contour Integrals

Given $J^*(A;B)$, an explicit formula for the n-correlation of zeros is obtained by integrating over carefully chosen contours. For a holomorphic, translation-invariant test function $f$: $$\sum_{0 < \gamma_1, ..., \gamma_n < T} f(\gamma_1, ..., \gamma_n) = \frac{1}{(2\pi i)^n} \int_{C}... \int_{C} J^*_E(-iz_1, ..., -iz_n)\ f(iz_1, ..., iz_n)\ dz_1...dz_n + O\left(T^{1/2+\epsilon}\right)$$ This contour formulation (as in Theorem 9) ensures avoidance of poles from the integrand, with the exact prescription detailed in the corresponding section of the paper.

The structure of $J^*_E$ contains not just the “main term” (often of determinantal or Pfaffian type, depending on symmetry), but also all secondary—arithmetic—corrections. These lower order corrections typically arise through prime sums and Euler-type products involving $z_p(x)$ and appear explicitly in the formulas for $Z(S,T;C,D)$.

4. Lower Order (Arithmetic) Terms and Explicit Prime-dependent Factors

The Ratios Conjecture, by virtue of being sensitive to the Euler product structure of the zeta-function, provides formulas that retain the impact of prime powers and arithmetic subtleties. In contrast to earlier approaches (such as the trace formula or Montgomery’s pair-correlation for limited Fourier support), the CFZ method systematically produces every lower order correction, including those reflecting diagonal as well as “off-diagonal” (swapped shift) interactions.

These corrections manifest as prime-dependent local factors ($z_p(x), Z_p, A_{p,0}$) that feature in the multiplicative component of $Z(S,T;C,D)$. This arithmetic sensitivity allows for precise matching with observed lower order terms in empirical statistics (e.g., in the work of Bogomolny–Keating) and substantially refines the link between zeta zeros and primes.

5. Comparison with Random Matrix Theory and Universality

The same methodology directly applies to averages over characteristic polynomials in $U(N)$. In the matrix context, one computes

$$R_{RMT}(A,B;C,D) = \left\langle \prod_{a\in A} P_X(e^{-a}) \bigg/ \prod_{b\in B} P_X(e^{-b}) \right\rangle_{U(N)}$$

and, after differentiation, analogous formulae to the zeta-case emerge—now without arithmetic complications. In the large-$N$ limit, as the integration variables are appropriately scaled, the local arithmetic content vanishes and the classical determinantal (sine kernel) predictions of RMT are recovered.

This structural similarity allows for direct term-by-term comparison of the arithmetic (zeta) and RMT cases, demonstrating that the observed universality of local spectral statistics (e.g., the Montgomery–Odlyzko law) is captured at all orders by the conjecture.

6. Impact and Applications

The systematic algebraic and analytic machinery of the Ratios Conjecture endows the n-correlation formulas with several key advantages:

• Full Lower-order Series: All arithmetic corrections are included systematically, not just the leading term.
• Unified Treatment: Works equally for the Riemann zeta function, other L-functions, and random matrix analogues.
• Generality in Test Functions: Allows for the use of general holomorphic, translation-invariant test functions, not only those with restricted support.
• Numerical Validation: Predictions for lower order terms have been confirmed by comparing with number-theoretical computations and with numerics.

When evaluated numerically or compared against rigorous calculations, the ratios-based predictions show excellent agreement for the full n-correlation, including both leading and lower order terms.

7. Implications and Limitations

The results derived from the Ratios Conjecture provide a comprehensive testing ground for conjectural links between primes, zeros, and random-matrix statistics. The inclusion of all arithmetic terms enables detailed investigations into universality and arithmetic-specific deviations. The framework overcomes key limitations of preceding methods—for example, those restricted to pair-correlation and to test functions of compact support.

However, the conjecture itself is not proven; while it consistently matches number-theoretical results and RMT predictions, it is grounded in a “formal recipe” rather than a rigorous derivation. Its validity hinges on deep interactions between the zeros of L-functions and the arithmetic content of their Euler products—a subject of ongoing research.

Feature	Traditional Approach	Ratios Conjecture Approach
Leading order term	Yes (determinantal/Pfaffian)	Yes
Systematic lower order corrections	Typically No	Yes, explicit
Arbitrary test function support	No (often restricted)	Yes
Random matrix, zeta and L-function families	Different treatments	Unified

In conclusion, the Ratios Conjecture of Conrey, Farmer, and Zirnbauer offers a unified and explicit methodology for predicting the n-correlation functions of zeros of the Riemann zeta function, including the full series of lower order arithmetic corrections, and ensures direct comparability between number-theoretical objects and random matrix ensembles (0803.2795).