One Level Density Conjecture

• The one level density conjecture defines the statistical distribution of normalized low-lying zeros in automorphic L-functions, matching eigenvalue patterns from random matrix theory.
• It employs techniques like the explicit formula and averaging over families to reveal the impact of symmetry and arithmetic on the main and lower-order terms.
• Recent studies indicate that family-specific secondary terms can alter the expected universality, challenging predictions from the Ratios Conjecture.

The one level density conjecture is a central statement in analytic number theory concerning the distribution of normalized low-lying zeros of families of automorphic $L$-functions. It predicts that the statistical density of zeros near the critical point $s = 1/2$ in large families of $L$-functions matches the density of eigenvalues near 1 in certain classical compact groups from random matrix theory (RMT), with symmetry type determined by the family. The conjecture is motivated by results of Katz and Sarnak and is now foundational in establishing connections between arithmetic statistics of families of $L$-functions and random matrix theory.

1. Rigorous Definition and Normalization

Given a natural family $\mathcal{F}$ of automorphic $L$-functions with analytic conductor $C(f)\to\infty$ for $f\in\mathcal{F}$, for each $L(s, f)\in\mathcal{F}$, denote the nontrivial zeros as $s = 1/2 + i\gamma_{f,j}$ (assuming the Generalized Riemann Hypothesis as needed). For an even Schwartz-class test function $\phi:\mathbb{R}\to\mathbb{R}$ with compactly supported Fourier transform $\widehat{\phi}$, the (scaled) one-level density is defined as

$$D_1(\mathcal{F};\phi) := \lim_{X\to\infty} \frac{1}{|\mathcal{F}(X)|} \sum_{f \in \mathcal{F}(X)} \sum_j \phi\left( \widetilde\gamma_{f,j}\right),$$

where $|\mathcal{F}(X)|$ counts elements in the family with $C(f)\le X$ and normalization

$$\widetilde\gamma_{f,j} = \frac{\gamma_{f,j}}{2\pi} \log C(f).$$

Variants employ smooth weights or restrict to families with additional arithmetic or symmetry properties. The normalization ensures the average spacing of zeros near the central point is $\sim1$.

2. The Katz–Sarnak Conjecture and Symmetry Types

Katz and Sarnak conjectured that for any natural family $\mathcal{F}$, as the analytic conductor grows, the limiting one-level density is given by the eigenvalue density of a classical compact group $G$: $$D_1(\mathcal{F};\phi) \xrightarrow{|\mathcal{F}|\to\infty} \int_{-\infty}^{\infty} \phi(x) W_G(x)\,dx,$$ where $W_G$ is the RMT scaling kernel corresponding to the relevant group:

• Unitary: $W_U(x)=1$,
• Symplectic: $W_{USp}(x)=1 - \frac{\sin(2\pi x)}{2\pi x}$,
• Orthogonal (even): $W_{SO^+}(x)=1 + \frac{\sin(2\pi x)}{2\pi x}$,
• Orthogonal: $W_O(x) = 1 + \tfrac{1}{2}\delta_0(x)$.

The symmetry type $G$ is determined by the nature of the family (e.g., self-duality, sign of functional equation) and underlies the universality principle that emerges in the statistics of low-lying zeros.

3. The Ratios Conjecture and Lower-Order Terms

Beyond the main RMT-predicted term, the $L$-function Ratios Conjecture of Conrey–Farmer–Zirnbauer provides a heuristic recipe for predicting lower-order terms in densities of low-lying zeros for $L$-function families. For suitably small support of $\widehat{\phi}$ and appropriate normalization, the Ratios Conjecture predicts, for example in the family of Dirichlet $L$-functions,

$$D_1(\phi; Q) = \widehat{\phi}(0) \left[1 - \frac{A_1}{\log Q}\right] + \int_0^\infty \frac{\widehat{\phi}(0) - \widehat{\phi}(t)}{Q^{t/2} - Q^{-t/2}}\,dt + O_\varepsilon(Q^{-1/2+\varepsilon}),$$

with

$$A_1 = \log(4\pi e^\gamma) + 1 + \sum_p \frac{\log p}{p(p-1)},$$

and explicit constants dependent on the family.

The conjecture further suggests optimal error exponents for these densities, typically $Q^{-1/2 + \varepsilon}$ in the Dirichlet case, and relates lower-order corrections to the arithmetic of the family and support of the test function.

4. Thresholds in Fourier Support and New Arithmetical Terms

Several studies establish that for restricted support of the test function's Fourier transform, one unconditionally recovers the main RMT-predicted term plus lower-order arithmetic corrections that can be calculated via the explicit formula. A key phenomenon is the emergence of new lower-order terms as the support is extended:

• For Dirichlet $L$-functions with test function $\phi$ satisfying $\operatorname{supp}\widehat{\phi} \subset [-1,1]$, all lower-order corrections are $O(Q^{\sigma/2-1})$ with $\sigma = \sup(\operatorname{supp}\widehat{\phi})$, and the random-matrix prediction plus an explicit correction is unconditionally verified, see Theorem A in (Fiorilli et al., 2011).
• Boh improvement to $\operatorname{supp}\widehat{\phi} \subset (-2,2]$ requires extra arithmetic input (stronger distribution of primes in arithmetic progressions), and for prime moduli, the Ratios Conjecture prediction is exact up to $O(Q^{-1/2+\varepsilon})$, with no new lower order (Fiorilli et al., 2011).
• For $\operatorname{supp}\widehat{\phi}\subset(-3/2,3/2)$, a genuinely new explicit arithmetical term of size $Q^{-1/2}/\log Q$ appears in the density, not predicted by the standard Ratios Conjecture, as shown in Theorem C of (Fiorilli et al., 2011). This establishes that the $Q^{-1/2}$ exponent for the error is best possible.

The analysis reveals a sharp transition at $\sigma=1$, where the nature and scaling of the lower-order terms change, mirroring analogous phenomena (e.g., in Montgomery's pair correlation).

5. Techniques: Explicit Formula and Prime Sums

The main technical engine in verifying the one level density conjecture and its refinements is Weil's explicit formula, which expresses sums over zeros of $L$-functions in terms of prime sums. For example, for a primitive Dirichlet character $\chi$,

$$\sum_\rho \phi\left(\frac{\gamma_\chi \log Q}{2\pi}\right) = \text{main term in } \phi(0) + \sum_{n=1}^\infty (\chi(n) + \overline{\chi}(n)) \frac{\Lambda(n)}{\sqrt{n}} \phi\left(\frac{\log n}{\log Q}\right) + \ldots$$

Averaging such expressions over the family and exploiting orthogonality, dispersion estimates, Linnik's method, and deep results on the distribution of primes in arithmetic progressions, researchers have unconditionally extended the admissible support beyond traditional “diagonal” bounds (e.g., up to $(-2-50/1093, 2+50/1093)$ in (Drappeau et al., 2020)).

When more extensive support is sought, progress depends on refined bounds for bilinear forms, Kloosterman sums, and prime distribution conjectures (e.g., Montgomery's conjecture). Under such conjectures, the random-matrix predictions for arbitrarily large finite support follow, always with error at least $Q^{-1/2 + o(1)}$.

6. Contrasts, Refined Conjectures, and Family-Dependent Phenomena

Recent work has demonstrated that the universality of lower-order terms can be broken. In the family of $L$-functions associated to non-Galois cubic Dedekind zeta functions, Cho–Fiorilli–Lee–Södergren (Cho et al., 2021) find a fractional power term $X^{-1/6}/\log X$ in the one-level density, directly linked to the secondary main term in the field count (Roberts' Conjecture). The standard Ratios Conjecture fails to predict this term; only by incorporating the “secondary” counting terms into the heuristic can the refined conjecture match rigorous results.

General lessons include:

• The random-matrix prediction typically describes the main term, but arithmetic and geometric structure of the family can produce non-universal lower-order terms, especially when the family is not harmonic or secondary terms enter the family size.
• The error terms in the Ratios Conjecture are not always $O_\varepsilon(X^{-1/2+\varepsilon})$; in special families with secondary main terms in the counting, this can be $O_\varepsilon(X^{-1/3+\varepsilon})$ (Cho et al., 2021).
• The transition in the support of $\widehat{\phi}$ at certain critical values is accompanied by abrupt changes in the size and structure of the lower-order (arithmetical) corrections.

7. Weighted One-Level Density and Extensions

The paper of one-level densities has recently been broadened to averaged statistics weighted by central or special values of $L$-functions (e.g., $|L(1/2, \chi)|^k$). Under these “tilted” measures, the main term in the density may change, exhibiting symmetry shifts and new universal kernels, such as the pair-correlation GUE kernel (Fazzari, 2021, Sugiyama et al., 2022). For instance, in Dirichlet families,

$$\lim_{q\to\infty} \frac{\sum_\chi |L(1/2, \chi)|^2 D(\chi, \phi)}{\sum_\chi |L(1/2, \chi)|^2} = \int_{-\infty}^{\infty} \phi(x) [1 - (\sin \pi x / \pi x)^2]\,dx$$

for suitably small support, confirming predictions of Fazzari and others.

Further, the comprehensive paper of weighted one-level densities and their kernels, alongside refined Ratios Conjectures, highlights a rich interplay between $L$-function arithmetic, zero statistics, and random matrix heuristics, leading to a more nuanced understanding of universality, symmetry, and lower-order phenomena in number theory.