One-Level Density of Low-Lying Zeros

• One-Level Density of Low-Lying Zeros is a statistic that measures the distribution of nontrivial zeros near the critical point in families of L-functions.
• The universal main term, dictated by symmetry types from random matrix theory, is derived using Fourier transforms and explicit formula methods.
• Lower-order terms reveal arithmetic fingerprints such as discriminants and class numbers, refining predictions beyond the universal behavior.

The one-level density of low-lying zeros is a central statistic in analytic number theory that quantifies the averaged distribution of nontrivial zeros of families of $L$-functions near the critical central point (typically $s=1/2$). It encodes deep relationships between arithmetic properties of the $L$-functions, their underlying symmetry type, and the universality classes suggested by random matrix theory (RMT). Rigorous theorems as well as conjectural predictions provide explicit asymptotic formulas for one-level densities, with particular attention to the “main term”—governed by the symmetry type—and arithmetic lower-order corrections specific to the family in question.

1. Definition and Formulation of One-Level Density

Let $\mathcal{F}$ denote a family of automorphic or number field $L$-functions, such as ideal class $L$-functions of a sequence of CM-fields, primitive Dirichlet $L$-functions, or Hecke $L$-functions attached to algebraic objects. For each $L$-function $L(s)$ in the family, under the Generalized Riemann Hypothesis the nontrivial zeros can be written as $1/2 + i\gamma_j$. For a real even Schwartz test function $\phi$, whose Fourier transform $\hat\phi$ has compact support, the one-level density is defined as: $$D(L; \phi) = \sum_j \phi\left( \gamma_j \frac{\log Q}{2\pi} \right)$$ where $Q$ is the analytic conductor associated with $L(s)$. The averaged (or “family”) density is then

$$\mathcal{D}_{\mathcal{F}}(\phi) = \lim_{Q \to \infty} \frac{1}{|\mathcal{F}(Q)|} \sum_{L \in \mathcal{F}(Q)} D(L; \phi)$$

where $\mathcal{F}(Q)$ is the subfamily with conductors near $Q$.

The choice of scaling by $\log Q$ ensures that the mean density of zeros is normalized for asymptotic comparison across the family.

2. Universal Main Term: Symmetry Types and Random Matrix Theory

For broad classes of families, the main term in the one-level density exhibits universality, being dictated by the symmetry type (unitary, symplectic, or orthogonal) as postulated in the Katz–Sarnak philosophy. For instance, for a symplectic symmetry type—the haLLMark of quadratic Dirichlet $L$-functions, ideal class group $L$-functions for imaginary quadratic fields, and Hecke $L$-functions for CM-fields over a fixed totally real field—the expected main term is: $\widehat\phi(0) - \frac{1}{2} \phi(0)$ This matches precisely the scaling limit of eigenvalues of random matrices from the compact symplectic group $\mathrm{USp}(2N)$ in the RMT analogy (Miller et al., 2010). More generally, the main term does not depend on arithmetic invariants of the family but rather on the global symmetry type, and remains robust under considerable generalization across families, such as extensions from imaginary quadratic to broader CM-fields over a fixed totally real subfield.

For unitary families, such as certain quartic Hecke $L$-functions or families parameterized by primitive characters (with odd weight) over $\mathbb{Q}(i)$, the main term is simply $\widehat\phi(0)$, the Fourier transform of the test function at the origin.

Orthogonal and other symmetry types produce distinct kernel functions in the main term, all of which are governed by the same universality principle from random matrix theory (see discussion in (Alpoge et al., 2013, Gao et al., 2018), and RMT surveys).

3. Lower-Order Terms and Emergence of Arithmetic Invariants

Although the main term is universal, the arithmetic of the underlying fields first appears in explicit lower-order correction terms. These corrections arise at order $1/\log Q$, where $Q$ is the conductor or discriminant parameter.

In families of CM-fields over a fixed totally real field $K_0$ (with class number $h$, discriminant $\Delta$, degree $N$), the expansion for the one-level density is: $$D_{\mathcal{C}_e(K)}(\phi) = \widehat\phi(0) - \frac12\phi(0) + \frac{1}{\log\Delta} \cdot A(K,\phi) + O\left( \frac{\log\log\Delta}{\log\Delta} \right)$$ where $A(K,\phi)$ contains arithmetic data, for example: $$A(K,\phi) = \phi(0)\left[ \log\left( \frac{\sqrt{\Delta}}{2} \right) - 2N\gamma_{\mathrm{EM}} - 2N\log 8 \right] + T(K_0, \ldots)$$ with $\gamma_{\mathrm{EM}}$ the Euler–Mascheroni constant, $N = [K_0:\mathbb{Q}]$, and $T(K_0, \cdots)$ an explicit arithmetic constant involving the residue of the Dedekind zeta function of $K_0$ (Miller et al., 2010).

This emergence of arithmetic information in the $1/\log\Delta$ term reflects the “breaking” of universality at the lower order level, offering a sensitive probe of the field's discriminant, class number, and other invariants.

In every rigorously analyzed family to date, this two-term structure—universal main term plus arithmetic-dependent correction—persists, revealing the subtle interplay of symmetry and arithmetic fingerprints.

4. Methodological Foundations: The Explicit Formula and Averaging

The analytic backbone is the explicit formula, relating weighted sums over zeros of $L$-functions to sums over primes via the Fourier transform of the chosen test function. The formula takes the schematic form: $$D(L; \phi) = \widehat\phi(0) (\log Q) - 2\sum_{n} \frac{\Lambda(n)}{\sqrt{n}\log Q} a(n) \widehat\phi\left( \frac{\log n}{\log Q} \right) + (\text{lower order Gamma and archimedean terms})$$ where $\Lambda(n)$ is the von Mangoldt function and $a(n)$ are coefficients encoding arithmetic, such as Hecke eigenvalues. The range of support of $\widehat\phi$ controls the length of the prime sum (with $(–1,1)$ typically being the “trivial” range; extending beyond this involves additional analytic challenges).

Averaging over the family (for instance, over class group characters or over fields with fixed discriminant) often enables the diagonalization of the main term and, when possible, permits extraction of the lower-order main term after careful parameter counting and Poisson summation (Miller et al., 2010).

The universality in the main term is a consequence of the diagonalization after averaging, while the arithmetic content of the lower-order terms is a result of known dependencies of the explicit formula coefficients on discriminant, zeta residues, and associated constants.

5. Random Matrix Theory and Symmetry Type Verification

The agreement of the main term of the one-level density with the kernel associated to a given symmetry class—unitary, symplectic, or orthogonal—provides strong evidence, both conjectural (Katz–Sarnak philosophy) and rigorous (where proven), that the low-lying zeros of these $L$-functions behave like the scaled eigenvalues of random matrices in the appropriate group.

For example, the appearance of

$\widehat\phi(0) - \frac12\phi(0)$

signals symplectic symmetry ($\mathrm{USp}$), whereas

$$\int_{-\infty}^{\infty} \phi(x)\, W_{\mathrm{O}}(x)\, dx$$

with $W_{\mathrm{O}}(x)$ the kernel for the orthogonal ensemble, signals orthogonal symmetry (Alpoge et al., 2013). This concordance persists as the family is enlarged from limited examples (such as imaginary quadratic fields [FI]) to broad sequences of CM-fields of growing discriminant and class number (Miller et al., 2010).

Moreover, the results show that universality in the main term is unaffected by arithmetic complication, but the lower-order terms—the “fine structure”—allow for distinguishing between families whose leading behavior is indistinguishable.

6. Broader Implications for Families of Number Field $L$-Functions

The methodology and results extend naturally to much broader classes of $L$-functions, including those of number field origin (e.g., Hecke $L$-functions for general sequences of CM-fields, quadratic and higher-degree twists, and fields with various class number and arithmetic properties). The extension does not require the field to possess any special feature, as long as the family is arranged with a meaningful symmetry type (such as all CM-extensions over a fixed totally real base).

In all these families, the structure

$$\text{universal main term (symmetry group)} + \text{arithmetic-dependent lower-order term}$$

represents a robust phenomenon; the main term predicts the global symmetry, while the detailed lower-order arithmetic reveals the “family fingerprint.”

This structural paradigm connects deep aspects of the theory of $L$-functions, random matrix theory, and the underlying field arithmetic, and provides the analytic backbone for modern results on non-vanishing at the central point and fine distinctions between closely related families.

7. Quantitative Results and Arithmetic “Fingerprinting”

Explicitly, for families of ideal class group $L$-functions over CM-fields $K$ of discriminant $\Delta$ (under appropriate assumptions, such as narrow class number one for the totally real subfield $K_0$), the one-level density expansion is: $$D_{\mathcal{C}_e(K)}(\phi) = \widehat\phi(0) - \frac12 \phi(0) + \frac{1}{\log\Delta}\left[\phi(0)\left( \log\left(\frac{\sqrt{\Delta}}{2}\right) - 2N\gamma_{\mathrm{EM}} - 2N\log8 \right) + T(K_0)\right] + O\left( \frac{\log\log \Delta}{\log\Delta} \right)$$ allowing, in principle, the extraction of explicit arithmetic features (e.g., discriminant-dependent constants, residues of zeta functions, Euler constants) from empirical statistics of zeros near the center.

This arithmetic “fingerprinting” is prevalent in all advanced studies of the one-level density for families of number field $L$-functions and underpins the “deviation from universality” for secondary terms, as now seen across broad aspects of automorphic and algebraic number theory (Miller et al., 2010).

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In summary, the one-level density of low-lying zeros is governed by a universal symmetry-dictated main term and sensitive arithmetic lower-order terms. This two-term structure, emerging from explicit formula methods and family averaging, captures both global universality (as modeled by random matrix ensembles) and family-specific arithmetic, providing a quantitative and conceptual bridge between algebraic invariants of fields and the statistical distribution of zeros of $L$-functions.