Baker-Montgomery Conjecture on L-Function Zeros

• Baker-Montgomery Conjecture is a hypothesis on the asymptotic behavior of real zeros of L'(s,χ₋d), predicting their increase at a rate proportional to log log |d|.
• The conjecture connects analytic number theory with random matrix theory by employing Selberg's short Dirichlet polynomial approximations and probabilistic models to estimate zero distributions.
• Recent work by Lamzouri and Nath has nearly achieved an unconditional resolution up to a polylogarithmic factor, highlighting both upper and lower bounds for real zeros in the critical interval.

The Baker-Montgomery conjecture concerns the asymptotic behavior of the number of real zeros of the derivative of quadratic Dirichlet $L$-functions, specifically $L'(s, \chi_d)$, within the critical interval $[1/2,1]$. Given its focus on fundamental discriminants and the distribution of zeros near the critical line, the conjecture is a central problem in the analytic study of $L$-functions and their derivatives, linking zero distribution phenomena to random matrix theory and sieving methods. Recent advances, notably by Lamzouri and Nath, have nearly resolved the conjecture unconditionally up to a polylogarithmic factor, with a full resolution conditional on well-motivated analytic hypotheses.

1. Formal Statement of the Baker–Montgomery Conjecture

Let $d$ be a fundamental discriminant and $\chi_d$ the associated primitive quadratic Dirichlet character. Define $R_d(\sigma_1,\sigma_2)$ as the number of real zeros of $L'(s,\chi_d)$ in the interval $[\sigma_1,\sigma_2]$. The conjecture posits:

For almost all fundamental discriminants $d \to \infty$,

$> R_d(1/2,1) \sim C \log\log|d| >$

for some positive constant $C$; in particular, $R_d(1/2,1) \asymp \log\log|d|$.

This predicts that the number of real zeros of $L'(s,\chi_d)$ in $[1/2,1]$ for almost all $d$ grows like the double logarithm of $|d|$.

2. Definitions and Foundational Objects

• Quadratic Dirichlet Character $\chi_d(n)$: The primitive character modulo $|d|$ defined via the Kronecker symbol $(d/n)$.
• Fundamental Discriminant $d$: The discriminant of the quadratic field $\mathbb{Q}(\sqrt{d})$, with $d \equiv 1 \pmod{4}$ and squarefree, or $d=4m$, $m\equiv2,3\pmod{4}$ and squarefree.
• Dirichlet $L$-function:

$$L(s,\chi_d) = \sum_{n=1}^\infty \frac{\chi_d(n)}{n^{s}},\qquad \Re s>1$$

Analytically continued to $\mathbb{C}$; its derivative is $L'(s, \chi_d) = \frac{d}{ds}L(s,\chi_d)$.

3. Unconditional Results: Upper and Lower Bounds

Upper Bound (Lamzouri–Nath):

Let $\nu(x)\to\infty$ as $x\to\infty$, and $D(x) = \{$fundamental discriminants $d$ in $(x/2, x]$ in a fixed progression$\}$, $|D(x)| \asymp x$. Theorem 1.1 asserts:

$$R_d\left(\frac{1}{2}+\frac{\nu(x)}{\log x}, 1\right) \ll (\log\log x) (\log\log\log x)$$

for almost all $d \in D(x)$. Thus, outside a negligible exceptional set, $L'(s,\chi_d)$ has at most $O((\log\log |d|)(\log\log\log |d|))$ real zeros in $[\frac{1}{2}+\frac{\nu(|d|)}{\log|d|}, 1]$.

Proof strategy summary:

• Approximate $-\frac{L'}{L}(s,\chi_d)$ by short Dirichlet polynomials (Selberg approximation).
• Develop a probabilistic random model by replacing $\chi_d(n)$ with i.i.d. random variables.
• Compare moments via discrepancy bounds.
• Use a discretized net $\{z_j = 1/2+3^{-j}\}$; control $L_d(z_j)$ uniformly.
• Apply Jensen’s formula over concentric circles to bound the count of real zeros.

Lower Bound (Klurman–Lamzouri–Munsch):

For almost all $d\in D(x)$,

$R_d(1/2,1) \gg \frac{\log\log|d|}{\log_4|d|}$

Moreover, all such zeros lie in $[1/2+1/(\log x)^{1/5},1]$.

Combined Order of Magnitude (Corollary 1.2):

For almost all $d$,

$$R_d\left(\frac{1}{2}+\frac{\nu(x)}{\log x},1\right) = (\log\log x)(\log\log\log x)^\theta$$

for some $|\theta|\leq 1$, exhibiting the optimal shape up to a polylogarithmic factor.

4. Conditional Results and Hypotheses

Assuming a mild zero-repulsion hypothesis on the low-lying zeros of quadratic Dirichlet $L$-functions—precisely, that for each $d\in D(x)$ there exists $\nu(x)\rightarrow\infty$ so that $L(s,\chi_d)$ has no zeros in a disk centered at $1/2+\nu(x)/\log x$ of suitable small radius (Hypothesis $L_d$)—a full resolution of the conjecture is obtained up to the $\log\log\log|d|$ factor:

Theorem 1.3 (Conditional): Assuming Hypothesis $L_d$ for almost all $d \in D(x)$,

• $R_d(1/2,1) \ll (\log\log x)(\log\log\log x)$
• $$R_d(1/2,\,1/2+\nu(x)/\log x) = o(R_d(1/2+\nu(x)/\log x, 1))$$

When GRH and the one-level density conjecture of Katz and Sarnak hold (established for support up to $(-2,2)$ by Özlük and Snyder), Hypothesis $L_d$ is satisfied for almost all $d$.

Key technical tools employed include:

• Orthogonality relations for $\chi_d(n)$ (Lemma 2.1)
• Zero-density theorems in the critical strip (Lemma 2.7)
• Selberg's short Dirichlet polynomial bounds (Lemma 3.1)
• Discrepancy bounds versus random models (Proposition 4.2)
• Moment estimates for $L(s)$ near and away from $1/2$ (Lemmas 4.3, 5.1, Proposition 5.2)

5. Error Terms, Polylogarithmic Factors, and Localization

Error Source and Estimates

• Exceptional set for Theorem 1.1: size $$O\left(x\cdot\exp(-c\nu(x)) + x \cdot (\log\log x)^{-3}\right)$$, hence for slowly diverging $\nu(x)$, nearly all $d$ are included.
• For Theorem 1.3: the exceptional proportion is $O(\sqrt{\log\nu(x)/\nu(x)})$.
• The $\log\log\log x$ factor originates from:
  • Multiple applications of Jensen’s formula on nets of points (with $J\sim\log\log x$), each incurring an additive $\log\log\log x$ term;
  • Discrepancy error terms near the boundary of the critical interval, scaling like $(V_j \log\log V_j/\log d)^{1/2}$, with $V_j = 3^j$.

Support for "All Zeros to the Right":

Under Hypothesis $L_d$, almost all real zeros of $L'(s, \chi_d)$ in $[1/2,1]$ satisfy

$$R_d(1/2,1/2+\nu/\log x) = o(R_d(1/2+\nu/\log x,1)),$$

hence asymptotically all such zeros are confined to $[1/2+\nu/\log x,1]$. Methods employ moment bounds on $L(s)$ near $s_0=1/2+\nu/\log x$, Jensen's formula on nested circles, hole-size control by the zero-free disk, and Cauchy-integral mean-value estimates.

6. Remaining Obstacles and Steps Toward a Full Unconditional Proof

Unconditional results leave open the “near-critical” interval $[1/2,1/2+\nu/\log |d|]$, offering no control over zeros in this sliver, and have a $O(\log\log\log|d|)$ slack from the predicted asymptotic. Removal of the last polylogarithmic factor and full control as $\sigma\searrow 1/2$ would require:

• Stronger uniform discrepancy estimates for $-\frac{L'}{L}(s,\chi_d)$ for $1/2 < \sigma < 1/2 + O(1/\log x)$,
• Sharper understanding of large-value behavior of $-\frac{L'}{L}$ very close to the $1/2$-line,
• Improved zero-density estimates within the near-critical strip.

As of Lamzouri and Nath’s work, methods reach within a polylogarithmic factor of confirming the conjecture, with a full conditional resolution when natural hypotheses on the distribution of low-lying zeros are assumed.

7. Broader Context and Related Developments

The Baker–Montgomery conjecture is part of a broader program investigating the vertical and horizontal distribution of zeros of $L$-functions and their derivatives. The moment-discrepancy methodology, as refined by Lamzouri and Nath, uses a random model to approximate $L$-function behavior and integrates zero-density, orthogonality, and polynomial approximation techniques. The work interacts with the one-level density conjecture and Katz–Sarnak random matrix models, providing strong evidence for deep connections between analytic multiplicities, random matrix statistics, and arithmetic geometry.

A plausible implication is that refinements of random-model comparisons and explicit moment computations for Dirichlet $L$-functions may generalize to higher-degree $L$-functions or to derivatives of automorphic $L$-functions, strengthening the bridge between analytic number theory and statistical mechanics models.

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Selected works:

• R. C. Baker & H. L. Montgomery, “Oscillations of quadratic L-functions,” 1989.
• O. Klurman, Y. Lamzouri & M. Munsch, “Sign changes of short character sums and real zeros of Fekete polynomials,” (Klurman et al., 4 Mar 2024).
• J. B. Conrey & K. Soundararajan, “Real zeros of quadratic Dirichlet L-functions,” Invent. Math. 150 (2002).
• A. E. Özlük & C. Snyder, “On the distribution of the nontrivial zeros of quadratic L-functions close to the real axis,” Acta Arith. 91 (1999).