Murmuration Phenomenon: Oscillation in Math & Biology

• Murmuration phenomenon is the emergent behavior observed in arithmetic statistics and bird flocking dynamics, characterized by oscillatory correlations and structured averaging.
• In arithmetic, rigorous trace formulas reveal persistent oscillatory biases in local automorphic coefficients and elliptic curve Frobenius traces that challenge naive independence heuristics.
• In biology, statistical-mechanical and agent-based models explain scale-free, coherent flocking patterns in birds, underpinning emergent murmuration dynamics.

Searching arXiv for papers on murmuration phenomena in both arithmetic and biological contexts. I’ll look up arXiv papers directly to support the article. The murmuration phenomenon designates two technically distinct but internally well-defined research objects. In arithmetic statistics and automorphic forms, it denotes a non-trivial oscillatory correlation between signs of functional equations and local coefficients such as Hecke eigenvalues or Frobenius traces, detected through conductor- or weight-windowed averages rather than vanishing under naive independence heuristics (Kuan et al., 15 Jul 2025). In collective-animal dynamics, it denotes the large-scale, coherent, rapidly reconfiguring flocking patterns of birds—especially starlings—whose organization is studied through topological interaction rules, hydrodynamic theories, and statistical-mechanical models (Bialek et al., 2011). The shared label reflects oscillatory, emergent structure, but the two literatures address different objects, methods, and explanatory aims.

1. Arithmetic definition and rigorous formulation

For a family $\mathcal F$ of self-dual automorphic representations $\pi$ with completed $L$-function

$$\Lambda(s,\pi)=N(\pi)^sG(s)L(s,\pi)=\varepsilon(\pi)N(\pi)^{1-s}G(1-s)L(1-s,\pi),$$

murmuration behavior is formulated through smoothed expectations of local coefficients $\lambda_\pi(p)$ over analytic-conductor windows. In the framework of trace-formula analysis, one says that $\mathcal F$ exhibits a murmuration for the $p$-th coefficient if

$$\mathbb E_{\pi\in\mathcal F}[\lambda_\pi(p);X] = M_\Phi(p/X)+\mathrm{Err}_\Phi(p,X),$$

with a leading real-valued function $M_\Phi$ and a strictly smaller error term (Lowry-Duda, 2 Jun 2025). This definition isolates a leading correlation between $\lambda_\pi(p)$ and the conductor $\pi$0.

The most explicit rigorous weight-aspect result in the material considered here concerns level-1 holomorphic cuspidal Hecke eigenforms. Let $\pi$1 be the set of Hecke-normalized newforms of level $\pi$2 and weight $\pi$3, with functional equation

$\pi$4

where $\pi$5 and, in level $\pi$6, $\pi$7. Writing $\pi$8 and using Petersson weights

$\pi$9

Kuan and Lesesvre study the ratio of a weighted numerator

$L$0

to the corresponding denominator

$L$1

The murmuration phenomenon is precisely the assertion that $L$2 tends to a non-zero oscillatory function of $L$3, rather than to $L$4 as would occur under absence of correlation between $L$5 and $L$6 (Kuan et al., 15 Jul 2025).

Under GRH for Dirichlet $L$7-functions, if

$L$8

with Gaussian window $L$9 and $$\Lambda(s,\pi)=N(\pi)^sG(s)L(s,\pi)=\varepsilon(\pi)N(\pi)^{1-s}G(1-s)L(1-s,\pi),$$0, Kuan–Lesesvre obtain

$$\Lambda(s,\pi)=N(\pi)^sG(s)L(s,\pi)=\varepsilon(\pi)N(\pi)^{1-s}G(1-s)L(1-s,\pi),$$1

hence

$$\Lambda(s,\pi)=N(\pi)^sG(s)L(s,\pi)=\varepsilon(\pi)N(\pi)^{1-s}G(1-s)L(1-s,\pi),$$2

Equivalently, a non-trivial correlation of size $$\Lambda(s,\pi)=N(\pi)^sG(s)L(s,\pi)=\varepsilon(\pi)N(\pi)^{1-s}G(1-s)L(1-s,\pi),$$3 is visible as soon as the weight window satisfies $$\Lambda(s,\pi)=N(\pi)^sG(s)L(s,\pi)=\varepsilon(\pi)N(\pi)^{1-s}G(1-s)L(1-s,\pi),$$4 (Kuan et al., 15 Jul 2025). The same source states that previous work of Bober et al. required the shorter window $$\Lambda(s,\pi)=N(\pi)^sG(s)L(s,\pi)=\varepsilon(\pi)N(\pi)^{1-s}G(1-s)L(1-s,\pi),$$5.

2. Trace-formula mechanisms

The arithmetic theory is organized around explicit trace formulas. In the Petersson-trace treatment of holomorphic forms, the denominator is evaluated with the $$\Lambda(s,\pi)=N(\pi)^sG(s)L(s,\pi)=\varepsilon(\pi)N(\pi)^{1-s}G(1-s)L(1-s,\pi),$$6 case of the Petersson formula, so that the diagonal $$\Lambda(s,\pi)=N(\pi)^sG(s)L(s,\pi)=\varepsilon(\pi)N(\pi)^{1-s}G(1-s)L(1-s,\pi),$$7-term supplies the main bulk $$\Lambda(s,\pi)=N(\pi)^sG(s)L(s,\pi)=\varepsilon(\pi)N(\pi)^{1-s}G(1-s)L(1-s,\pi),$$8, while the Kloosterman-sum contribution is $$\Lambda(s,\pi)=N(\pi)^sG(s)L(s,\pi)=\varepsilon(\pi)N(\pi)^{1-s}G(1-s)L(1-s,\pi),$$9 (Kuan et al., 15 Jul 2025). For the numerator, the $\lambda_\pi(p)$0 case leaves only the off-diagonal:

$\lambda_\pi(p)$1

A decisive step is the transformation of the $\lambda_\pi(p)$2-sum into an oscillatory kernel. Splitting $\lambda_\pi(p)$3 and using Poisson summation in $\lambda_\pi(p)$4 together with the integral representation of $\lambda_\pi(p)$5, one finds

$\lambda_\pi(p)$6

with explicit sine-integrals $\lambda_\pi(p)$7 of $\lambda_\pi(p)$8. Weighting by $\lambda_\pi(p)$9 cancels the $\mathcal F$0 term and leaves exactly $\mathcal F$1, identified there as “the murmuration kernel” (Kuan et al., 15 Jul 2025). After the $\mathcal F$2-sum, the numerator becomes

$\mathcal F$3

Under GRH, the prime sum is replaced by

$\mathcal F$4

and the resulting $\mathcal F$5-sum is analyzed through the Dirichlet series

$\mathcal F$6

Mellin inversion on $\mathcal F$7 isolates the simple pole of $\mathcal F$8 at $\mathcal F$9, and only the leading term $p$0 in Li’s expansion of $p$1 contributes to the main term (Kuan et al., 15 Jul 2025).

A broader synthesis places these results within a common trace-formula template. Lowry-Duda emphasizes that known rigorous $p$2 murmurations for holomorphic and Maass families arise from Selberg, Petersson, or Kuznetsov trace formulas, with the geometric side yielding explicit series that determine the leading murmuration density (Lowry-Duda, 2 Jun 2025). The same note presents the weight-aspect theorem for $p$3 and the eigenvalue-aspect analogue for level-1 Hecke–Maass cuspforms, both of the form

$p$4

with the same density function $p$5 (Lowry-Duda, 2 Jun 2025). This suggests that the phenomenon is not an artifact of a single trace formula; in fact, Kuan–Lesesvre describe their work as the first approach using a relative trace formula, “showing its robustness” (Kuan et al., 15 Jul 2025).

3. Elliptic-curve murmurations and AI-assisted discovery

For elliptic curves over $p$6, the central local quantity is the Frobenius trace

$p$7

Fixing a conductor window $p$8 and a parity $p$9, He, Lee, Oliver, and Pozdnyakov define

$$\mathbb E_{\pi\in\mathcal F}[\lambda_\pi(p);X] = M_\Phi(p/X)+\mathrm{Err}_\Phi(p,X),$$0

and the murmuration

$$\mathbb E_{\pi\in\mathcal F}[\lambda_\pi(p);X] = M_\Phi(p/X)+\mathrm{Err}_\Phi(p,X),$$1

Empirically, as $$\mathbb E_{\pi\in\mathcal F}[\lambda_\pi(p);X] = M_\Phi(p/X)+\mathrm{Err}_\Phi(p,X),$$2 varies, the graph of $$\mathbb E_{\pi\in\mathcal F}[\lambda_\pi(p);X] = M_\Phi(p/X)+\mathrm{Err}_\Phi(p,X),$$3 oscillates about the horizontal axis with a persistent pattern of peaks and troughs, and these oscillations are described there as scale-invariant in the sense that the same wave-form reappears for disjoint intervals $$\mathbb E_{\pi\in\mathcal F}[\lambda_\pi(p);X] = M_\Phi(p/X)+\mathrm{Err}_\Phi(p,X),$$4 of very different sizes (He et al., 10 Mar 2026).

The same source makes machine-learning interpretability part of the mathematical narrative. Representing each curve by

$$\mathbb E_{\pi\in\mathcal F}[\lambda_\pi(p);X] = M_\Phi(p/X)+\mathrm{Err}_\Phi(p,X),$$5

PCA produces a first principal vector whose coordinates, plotted against the primes, reproduce the same oscillatory pattern as $$\mathbb E_{\pi\in\mathcal F}[\lambda_\pi(p);X] = M_\Phi(p/X)+\mathrm{Err}_\Phi(p,X),$$6. In supervised prediction of rank parity, saliency curves

$$\mathbb E_{\pi\in\mathcal F}[\lambda_\pi(p);X] = M_\Phi(p/X)+\mathrm{Err}_\Phi(p,X),$$7

again reveal the murmuration wave-form, and one-dimensional convolutional networks learn short kernels with alternating signs and magnitudes that match local features of the pattern (He et al., 10 Mar 2026).

The arithmetic interpretation is tied explicitly to Birch–Swinnerton-Dyer and random matrix theory. Since

$$\mathbb E_{\pi\in\mathcal F}[\lambda_\pi(p);X] = M_\Phi(p/X)+\mathrm{Err}_\Phi(p,X),$$8

the average of $$\mathbb E_{\pi\in\mathcal F}[\lambda_\pi(p);X] = M_\Phi(p/X)+\mathrm{Err}_\Phi(p,X),$$9 over parity-constrained families heuristically encodes rank information, and differentiating in $M_\Phi$0 recovers the murmuration curve $M_\Phi$1 (He et al., 10 Mar 2026). Under GRH for the family $M_\Phi$2, the same source states a first-order bias formula

$M_\Phi$3

which explains both the $M_\Phi$4 decay and the oscillatory phase.

A separate line of work studies elliptic curves ordered by naive height rather than conductor. For minimal models

$M_\Phi$5

Sawin and Sutherland consider

$M_\Phi$6

They conjecture that $M_\Phi$7 tends to $M_\Phi$8, where $M_\Phi$9 is given by an explicit double Bessel sum, and they prove a smoothed variant via the Voronoi summation formula (Sawin et al., 16 Apr 2025). The paper describes this as the first work to give an explicit formula for the murmuration density of a family of elliptic curves, in any ordering.

4. Extensions to higher-rank zeta functions

The arithmetic literature also contains a higher-rank generalization. For an elliptic curve $\lambda_\pi(p)$0, Weng’s non-abelian rank-$\lambda_\pi(p)$1 zeta function is

$\lambda_\pi(p)$2

with

$\lambda_\pi(p)$3

Shi and Weng define rank-$\lambda_\pi(p)$4 averages $\lambda_\pi(p)$5 by suitable linear normalizations of the coefficients $\lambda_\pi(p)$6, and report that the scatter-plots $\lambda_\pi(p)$7 show exactly the same murmuration patterns, in the same conductor ranges and arithmetic rank, as for $\lambda_\pi(p)$8 (Shi et al., 2024).

The theoretical explanation is an asymptotic expansion valid for $\lambda_\pi(p)$9:

$\pi$00

After the normalization used in the definition of $\pi$01, this yields

$\pi$02

so that, to leading order, $\pi$03 (Shi et al., 2024). In the same paper, a high-rank Sato–Tate statement is formulated via a renormalized invariant $\pi$04 whose limiting law is again the classical $\pi$05 density.

This suggests that the murmuration pattern is stable under passage from the classical Hasse–Weil setting to Weng’s higher-rank non-abelian zetas. A plausible implication is that the phenomenon is controlled less by the superficial form of local coefficients than by the leading asymptotic relation between the higher-rank coefficients and the classical Frobenius trace.

5. Biological murmurations: structure, density, and interaction topology

In ethology and statistical physics, a murmuration is the collective motion of a flock, especially of starlings, characterized by strong polarization, coherent turning, and long-ranged correlations. A foundational statistical-mechanical account models a flock snapshot by unit velocity vectors

$\pi$06

with maximum-entropy distribution

$\pi$07

equivalent to a classical Heisenberg model with effective Hamiltonian

$\pi$08

Assuming equal interaction with the $\pi$09 nearest neighbors, inference over 21 flocking events gives $\pi$10 and $\pi$11, independent of flock size or density; the correlation length $\pi$12 is observed to scale linearly with flock size $\pi$13 (Bialek et al., 2011). The same source attributes this scale invariance to Goldstone modes arising from spontaneous breaking of continuous rotational symmetry.

A distinct empirical issue is the inhomogeneous density profile. Lewis and Turner formulate a strictly metric-free model in which neighbors are defined topologically via the Delaunay triangulation, and birds are organized into topological shells: shell $\pi$14 is the convex hull, shell $\pi$15 consists of birds connected to shell $\pi$16, and so on (Lewis et al., 2019). Each bird combines coalignment

$\pi$17

with a shell-dependent “bounding” term built from shell-mates

$\pi$18

through

$\pi$19

followed by additive directional noise of strength $\pi$20 (Lewis et al., 2019).

Calibrated to starling data with $\pi$21 birds and $\pi$22, the best-fit parameters are

$\pi$23

The resulting density profile falls approximately linearly from $\pi$24 at the border to $\pi$25 at the center, in agreement with field data showing higher density at the border than near the centre (Lewis et al., 2019). The same work argues that birds on the edge require inward bias, whereas interior birds require outward bias, overturning the naive expectation that all individuals should move toward the center.

The model also proposes a mechanism for estimating topological depth from visual anisotropy. In a spherical flock of radius $\pi$26, a bird at radius fraction $\pi$27 has anisotropy

$\pi$28

and simulation indicates an almost linear correspondence $\pi$29, hence a monotonic map $\pi$30 (Lewis et al., 2019). This suggests a biologically local route from visual input to shell-dependent steering.

6. Dynamical theories of propagation, turning, and criticality

Several complementary models address how information and directional changes propagate across bird flocks. In “Flocking at the edge of chaos,” birds are self-propelled particles in two dimensions with Rayleigh–Helmholtz self-propulsion and topological neighbor sets defined relative to the center of mass. Attraction acts toward the $\pi$31 nearest neighbors in the “toward-COM” half-space, while repulsion acts on up to $\pi$32 neighbors in the opposite half-space (Bhattacharya et al., 2015). The continual reshuffling of these topological neighbor sets produces deterministic broadband “internal noise” without any stochastic forcing. Numerically, the velocity-fluctuation correlation length satisfies $\pi$33, the large-$\pi$34 correlation behaves as $\pi$35 with $\pi$36, the power spectral density obeys $\pi$37 with $\pi$38, and for $\pi$39 the largest Lyapunov exponent is $\pi$40 (Bhattacharya et al., 2015). The authors interpret this combination as evidence for self-organized criticality.

At hydrodynamic scale, “Silent Flocks” augments Toner–Tu velocity-density equations with an inertial spin field $\pi$41:

$\pi$42

$\pi$43

$\pi$44

Linearization yields a cubic dispersion relation with two characteristic mode speeds,

$\pi$45

corresponding to first sound and second sound (Cavagna et al., 2014). When $\pi$46, a finite $\pi$47-interval contains only damped modes, implying a “silent” size window

$\pi$48

in which neither density nor orientational disturbances can propagate coherently across the flock (Cavagna et al., 2014). Using reported starling values $\pi$49–$\pi$50 m/s and $\pi$51–$\pi$52 s, together with $\pi$53 m/s, the same paper places typical mid-sized flocks $\pi$54–$\pi$55 m in this window.

Agent-based models of spontaneous turning supply a different mechanism. Cristiani et al. introduce a delayed second-order system in which every bird is a potential turn initiator: each bird independently becomes a temporary leader with probability $\pi$56 per step, remains leader for at most $\pi$57 or until its nearest-neighbor distance exceeds $\pi$58, and then enters a refractory period $\pi$59 (Cristiani et al., 2020). With $\pi$60 topological nearest neighbors and explicit Euler time step $\pi$61, the model produces spontaneous, boundary-triggered turning cascades without permanent leaders or external stimuli.

Finally, Flock2 replaces vector-valued “social forces” by desired yaw and pitch turn rates computed from avoidance, alignment, cohesion, and peripheral boundary terms in each bird’s local frame (Hoetzlein, 2024). This orientation-based construction is coupled to an aerodynamic flight model with lift, drag, thrust, gravity, and quaternion-based body reorientation. In comparative simulations against Reynolds’ model, the average turning power is reported as $\pi$62 W for Flock2 and $\pi$63 W for Reynolds, while total mechanical power remains similar at $\pi$64 W and $\pi$65 W, respectively (Hoetzlein, 2024). The same source reports spontaneous orientation waves with periods in the $\pi$66–$\pi$67 s range, corresponding to $\pi$68–$\pi$69 Hz components, and gives a wave-speed estimate of approximately $\pi$70 m/s for a $\pi$71 m flock traversed in about $\pi$72 s.

7. Conceptual issues and open directions

In arithmetic, the main conceptual shift is from global averages expected to vanish to structured local averages that retain oscillatory bias after conditioning by parity, root number, or conductor window. Trace-formula methods show that these biases can be extracted as explicit geometric-side contributions rather than treated as numerical anomalies (Lowry-Duda, 2 Jun 2025). Current extensions point toward further families, including general-level Maass forms, symmetric powers, half-integral weight forms, Dirichlet characters, and quadratic twists (Lowry-Duda, 2 Jun 2025).

In biology, a persistent issue is which mechanism best explains scale-free collective response. The maximum-entropy account emphasizes Goldstone-mode propagation in highly polarized flocks (Bialek et al., 2011), whereas the edge-of-chaos model emphasizes self-organized criticality generated by deterministic topological reshuffling (Bhattacharya et al., 2015). The sources considered here do not resolve that contrast. Likewise, the distributed-bias model explains edge-crowding through outward bulk bias (Lewis et al., 2019), while hydrodynamic and agent-based models focus on propagation speeds, turning waves, delays, and transient leadership [(Cavagna et al., 2014); (Cristiani et al., 2020)].

Across both literatures, the term “murmuration” therefore denotes an emergent, structured pattern visible only after the correct coarse-graining. In arithmetic, that coarse-graining is averaging over automorphic or elliptic-curve families in the conductor, weight, or height aspect. In flocking theory, it is the passage from individual steering rules to collective density, correlation, and wave-propagation observables. A plausible implication is that the term has become attached not merely to oscillation, but to oscillation stabilized by a family-level averaging principle.