Simple and distinct zeros in a prime-modulus Dirichlet family at mesoscopic shrinking height

Abstract: Fix $0<α<1$. Let $q$ tend to infinity through odd primes, put $Q=\log q$, and set $I=(Q<sup>{-α},2Q<sup>{-α}]$. Averaging without weights over the $q-2$ nonprincipal characters modulo $q$, let $\mathcal N_q$ be the number of nontrivial zeros in $I$, counted with multiplicity; let $\mathcal N<sup>s_{0,q}$ and $\mathcal N<sup>*_{0,q}$ count, respectively, simple and distinct critical-line zero points; and let $\mathcal N_{d,q}$ count all distinct zero points, without a location restriction. We prove unconditionally that [ \mathcal N_q=\frac{q(\log q)^{1-α}}{2π}{1+o_α(1)}, ] [ \liminf_{q\to\infty}\frac{\mathcal N^s_{0,q}}{\mathcal N_q} \ge C_{\mathrm{MT}},\qquad \liminf_{q\to\infty}\frac{\mathcal N^*_{0,q}}{\mathcal N_q} \ge C_{\mathrm{MT}},\qquad \liminf_{q\to\infty}\frac{\mathcal N_{d,q}}{\mathcal N_q} \ge C_d, ] where [ C_{\mathrm{MT}}=\frac32-\frac1{\sqrt2}\cot\left(\frac1{\sqrt2}\right) =0.672500703679\ldots, \qquad C_d=\frac{1+C_{\mathrm{MT}}}{2}=0.836250351839\ldots . ] The physical height tends to zero, whereas $I$ contains $\asymp(\log q)<sup>{1-α}$ local mean spacings. The proof combines small-height counting and zero-density deletion with a finite smooth Gabor compression of Weil's Hermitian form. First and second matrix moments, together with an inertia-based rank--trace inequality, yield the three counting bounds. No form of the generalized Riemann hypothesis is assumed.

• The paper proves that, for every fixed 0<α<1, at least 67.25% of zeros in a mesoscopic shrinking window are simple critical-line zeros, while 83.63% occupy distinct zero points overall.
• It combines low-height zero counts, zero-density estimates, finite Gabor-frame compression of Weil’s form, family moments, and an inertia-based rank–trace inequality without assuming the generalized Riemann hypothesis.
• The results apply to averages over all nonprincipal characters modulo a prime q, with approximately q(log q)^(1−α)/(2π) zeros in the window, while the endpoint α=1 remains unresolved because key error terms and the sampled dimension no longer diminish or diverge.

The setting and the main theorem

This paper establishes unconditional lower bounds for the proportion of simple and distinct zeros of Dirichlet $L$-functions in a height window that shrinks to zero in absolute terms while containing a diverging number of local mean spacings. Fix $0<\alpha<1$, let $q\to\infty$ through odd primes, put $Q=\log q$, and consider the interval $I=(Q^{-\alpha},2Q^{-\alpha}]$. In the normalized coordinate $x=Q\gamma/(2\pi)$, this interval has length $\asymp Q^{1-\alpha}$, so it contains $\asymp(\log q)^{1-\alpha}$ mean spacings even though its physical ordinates tend to zero. Averaging without weights over the $q-2$ nonprincipal characters modulo $q$, the paper proves three family-level statements (2608.16034):

• $0<\alpha<1$0, where $0<\alpha<1$1 counts all nontrivial zeros in $0<\alpha<1$2 with multiplicity;
• $0<\alpha<1$3 and $0<\alpha<1$4, where $0<\alpha<1$5 counts simple critical-line zeros and $0<\alpha<1$6 counts distinct critical-line zero points;
• $0<\alpha<1$7, where $0<\alpha<1$8 counts all distinct zero points with no location restriction.

The constants are

$0<\alpha<1$9

Two features distinguish these results from prior work on simple zeros of Dirichlet $q\to\infty$0-functions, such as pair-correlation and Levinson-type treatments [Chandee–Lee–Liu–Radziwiłł 2014; Conrey–Iwaniec–Soundararajan 2013; Wu 2016; Sono 2025]. First, they are unconditional: no generalized Riemann hypothesis is assumed anywhere. Second, they hold at shrinking height, a regime where classical low-lying techniques do not directly apply because the window is far below any fixed positive ordinate. The bounds are averaged over the full character family of one prime modulus; no assertion about an individual character follows.

Method overview

The proof combines four inputs, each established separately before assembly:

Denominator and exceptional set. The paper uses the Hiary–Zhao symmetric counting formula, averaged argument bound ($q\to\infty$1 for $q\to\infty$2), and zero-density estimate. An endpoint-safe extraction lemma shifts both endpoints of $q\to\infty$3 by a small $q\to\infty$4 chosen to avoid all zero ordinates, so no half-weight convention is needed. Family symmetry under $q\to\infty$5 then identifies the positive-ordinate count with half the symmetric count, yielding $q\to\infty$6. The two normalized remainders are $q\to\infty$7 and $q\to\infty$8, both vanishing precisely for fixed $q\to\infty$9 — this is where the restriction $Q=\log q$0 first enters quantitatively.

A "bad" character is one possessing a zero with $Q=\log q$1 and $Q=\log q$2. Applying the Hiary–Zhao density estimate over logarithmically spaced shells in the real part, with the interval-length factor kept outside the power of $Q=\log q$3, gives $Q=\log q$4. This deletion bound is sharp enough that exceptional characters contribute negligibly to every subsequent ratio.

Gabor compression of Weil's form. For a smooth compactly supported window $Q=\log q$5 with $Q=\log q$6, the paper builds a finite Gabor frame: translates $Q=\log q$7 on a lattice of spacing $Q=\log q$8, restricted to centres $Q=\log q$9 in a slightly shrunken interior $I=(Q^{-\alpha},2Q^{-\alpha}]$0, giving dimension $I=(Q^{-\alpha},2Q^{-\alpha}]$1. The matrix $I=(Q^{-\alpha},2Q^{-\alpha}]$2 is the polarized Weil form evaluated on these test functions. A Poisson summation argument shows that the infinite lattice reproduces $I=(Q^{-\alpha},2Q^{-\alpha}]$3 exactly (only the zero dual mode survives, since $I=(Q^{-\alpha},2Q^{-\alpha}]$4), and finite-centre replacement errors are controlled by Schwartz decay envelopes plus exact character orthogonality, which holds because the prime cutoff $I=(Q^{-\alpha},2Q^{-\alpha}]$5.

First and second moments. The first moment evaluates cleanly:

$I=(Q^{-\alpha},2Q^{-\alpha}]$6

The prime side of the second moment splits into four orthogonality classes. Ratio terms force $I=(Q^{-\alpha},2Q^{-\alpha}]$7 and produce the weighted sum $I=(Q^{-\alpha},2Q^{-\alpha}]$8, evaluated via explicit PNT estimates (Johnston–Yang) as $I=(Q^{-\alpha},2Q^{-\alpha}]$9, where $x=Q\gamma/(2\pi)$0. Same-sign terms require $x=Q\gamma/(2\pi)$1 with $x=Q\gamma/(2\pi)$2; the oscillatory integral over the common translate is bounded by $x=Q\gamma/(2\pi)$3, killing them. Mixed archimedean–prime terms vanish by full-family orthogonality, and principal-character bookkeeping terms are $x=Q\gamma/(2\pi)$4. The result is

$x=Q\gamma/(2\pi)$5

Inertia certificate and exterior tail. On the zero side, a critical-line zero of multiplicity $x=Q\gamma/(2\pi)$6 contributes a rank-one positive block $x=Q\gamma/(2\pi)$7, while an off-line functional-equation orbit contributes a Hermitian block with Sylvester index $x=Q\gamma/(2\pi)$8. A self-contained rank–trace inequality — proved via a Ky Fan prefix inequality and eigenvalue-by-eigenvalue comparison — yields, for $x=Q\gamma/(2\pi)$9,

$\asymp Q^{1-\alpha}$0

with the same bound serving distinct critical-line points. The exterior zero contribution $\asymp Q^{1-\alpha}$1 (ordinates outside $\asymp Q^{1-\alpha}$2) is removed in trace norm using individual Fourier decay of the sampled vectors: for good characters, $\asymp Q^{1-\alpha}$3 for any preassigned $\asymp Q^{1-\alpha}$4, with the near-line regime requiring $\asymp Q^{1-\alpha}$5 and the far-line regime exploiting goodness via $\asymp Q^{1-\alpha}$6. One fixed integer Schwartz order $\asymp Q^{1-\alpha}$7 satisfies both tail conditions simultaneously because $\asymp Q^{1-\alpha}$8 is fixed.

Variational optimization and the Montgomery–Taylor constant

The certificate coefficient depends on the window only through $\asymp Q^{1-\alpha}$9. With $\asymp(\log q)^{1-\alpha}$0, minimizing $\asymp(\log q)^{1-\alpha}$1 leads to the Euler equation $\asymp(\log q)^{1-\alpha}$2, whose even solution is $\asymp(\log q)^{1-\alpha}$3. Global optimality follows from the identity $\asymp(\log q)^{1-\alpha}$4 for zero-mean $\asymp(\log q)^{1-\alpha}$5 together with an internal Dirichlet Wirtinger inequality, giving strict conditional convexity for $\asymp(\log q)^{1-\alpha}$6. Since $\asymp(\log q)^{1-\alpha}$7 is not compactly supported, admissibility is restored by smooth cutoffs converging to it. This yields

$\asymp(\log q)^{1-\alpha}$8

and taking $\asymp(\log q)^{1-\alpha}$9 after the fixed-$q-2$0 statement gives $q-2$1 — the classical Montgomery–Taylor constant, here arising from a finite Gabor window rather than pair correlation. The distinct-point bound $q-2$2 requires no new analysis; it is the exact combinatorial consequence that a multiple critical-line point contributes once to $q-2$3 but its orbit contributes two points to $q-2$4.

Limitations and open questions

The paper is explicit about the boundary of its method. All error estimates depend on the fixed auxiliary data, and none is uniform as $q-2$5, $q-2$6, or $q-2$7; these limits are taken only after the corresponding fixed-parameter liminf is proved. The endpoint $q-2$8 fails for structural reasons: the main family scale drops to $q-2$9, so the $q$0 averaged-argument remainder and several normalized $q$1 moment errors cease to be little-oh; moreover the sampled dimension $q$2 no longer diverges, and the near-line shell estimate loses decay regardless of how large the fixed Schwartz order $q$3 is taken. The authors state plainly that their analysis neither proves nor disproves a corresponding fixed-normalized-window theorem at $q$4. Two further caveats bear on interpretation: the results are purely family-level, and the proof relies on the Hiary–Zhao low-height input and density estimate as external hypotheses, so any refinement of those estimates would propagate directly into this framework.

Conclusion

For every fixed $q$5, the mesoscopic window $q$6 contains $q$7 zeros of the prime-modulus Dirichlet family counted with multiplicity, of which at least the fraction $q$8 are simple critical-line zeros (and the same fraction of multiplicity-normalized mass sits at distinct critical-line points), while at least $q$9 of the mass sits at distinct zero points overall. The argument is unconditional, combines a low-height count and zero-density deletion with finite Gabor moments, trace-norm localization, and an inertia-based rank–trace certificate, and leaves the endpoint $0<\alpha<1$00 as a clearly delineated open boundary of the method.