--- title: 'Murmuration Phenomenon: Oscillation in Math & Biology' url: https://www.emergentmind.com/topics/murmuration-phenomenon type: topic --- # Murmuration Phenomenon: Oscillation in Math & Biology Searching arXiv for recent 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 [2507.11418]. 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 [1107.0604]. 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 [2506.01640]. This definition isolates a leading correlation between $\lambda_\pi(p)$ and the conductor $N(\pi)$. The most explicit rigorous weight-aspect result in the material considered here concerns level-1 holomorphic cuspidal Hecke eigenforms. Let $H_k$ be the set of Hecke-normalized newforms of level $1$ and weight $k$, with functional equation $$ \Lambda(f,s)=\varepsilon_f\,\Lambda(f,k-s), $$ where $\varepsilon_f\in\{\pm1\}$ and, in level $1$, $\varepsilon_f=i^k$. Writing $a_f(p)=\lambda_f(p)$ and using Petersson weights $$ \omega_f=\left(\frac{(4\pi)^{k-1}}{\Gamma(k-1)}\right)^{1/2}\|f\|^{-1}, $$ Kuan and Lesesvre study the ratio of a weighted numerator $$ N=\sum_{p/K^2\in[A,B]}(\log p)\sum_{|k-K|\lesssim M}h(k/K)\sum_{f\in H_k}\omega_f\,\varepsilon_f\,\lambda_f(p) $$ to the corresponding denominator $$ D=\sum_{p/K^2\in[A,B]}(\log p)\sum_{|k-K|\lesssim M}h(k/K)\sum_{f\in H_k}\omega_f. $$ The murmuration phenomenon is precisely the assertion that $N/D$ tends to a non-zero oscillatory function of $K$, rather than to $0$ as would occur under absence of correlation between $\varepsilon_f$ and $\lambda_f(p)$ [2507.11418]. Under GRH for Dirichlet $L$-functions, if $$ K^{1/3+\epsilon}\ll M\ll K^{1-\epsilon}, $$ with Gaussian window $h(k)=\exp(-(k-K)^2/M^2)$ and $E=[A,B]$, Kuan–Lesesvre obtain $$ D\approx K^3|E|\widehat h(0),\qquad N\asymp K^2(\sqrt B-\sqrt A)\widehat h(0), $$ hence $$ \frac ND\sim K^{-1}(\sqrt B-\sqrt A). $$ Equivalently, a non-trivial correlation of size $\sim K^{-1}$ is visible as soon as the weight window satisfies $M\gg K^{1/3+\epsilon}$ [2507.11418]. The same source states that previous work of Bober et al. required the shorter window $M\gg K^{5/6+\epsilon}$. ## 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 $b=0$ case of the Petersson formula, so that the diagonal $\delta$-term supplies the main bulk $\sim K^3|E|\widehat h(0)$, while the Kloosterman-sum contribution is $\ll K^2|E|$ [2507.11418]. For the numerator, the $b=1$ case leaves only the off-diagonal: $$ N = 2\pi\sum_{p/K^2\in E}\log p\sum_k h(k/K)i^{-k}\sum_{c\ge1}\frac{S(1,p;c)}{c}\, J_{k-1}(4\pi\sqrt p/c). $$ A decisive step is the transformation of the $k$-sum into an oscillatory kernel. Splitting $k\bmod 4$ and using Poisson summation in $k$ together with the integral representation of $J_\ell$, one finds $$ S_a(x):=\sum_{k\equiv a(4)}u(k-1)J_{k-1}(x) = \frac{i}{2}V_2(x)+\frac{(-1)^{a/2+1}}{2}V_1(x), $$ with explicit sine-integrals $V_1,V_2$ of $\widehat u$. Weighting by $\varepsilon_f=i^k$ cancels the $V_1$ term and leaves exactly $V_2(x)$, identified there as “the murmuration kernel” [2507.11418]. After the $k$-sum, the numerator becomes $$ N = 2\pi i\sum_{c\ge1}\frac1c\sum_{p/K^2\in E}S(1,p;c)\log p\cdot V_2(4\pi\sqrt p/c). $$ Under GRH, the prime sum is replaced by $$ \sum_{p\le x,(p,c)=1}S(1,p;c)\log p = x\frac{\mu(c)^2}{\phi(c)}+O\!\left(\phi(c)x^{1/2}\log^2(cx)\right), $$ and the resulting $c$-sum is analyzed through the Dirichlet series $$ L(s)=\sum \frac{\mu(c)^2}{\phi(c)c^s}. $$ Mellin inversion on $V_2$ isolates the simple pole of $L(s)$ at $s=0$, and only the leading term $l=0$ in Li’s expansion of $V_2$ contributes to the main term [2507.11418]. A broader synthesis places these results within a common trace-formula template. Lowry-Duda emphasizes that known rigorous $\GL(2)$ 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 [2506.01640]. The same note presents the weight-aspect theorem for $S_k^\pm$ and the eigenvalue-aspect analogue for level-1 Hecke–Maass cuspforms, both of the form $$ \frac{\sum \log p\sum \lambda(p)}{\sum \log p\sum 1} = \pm \frac{1}{\sqrt{N}}\left(\frac{\nu(E)}{|E|}+o_E(1)\right), $$ with the same density function $\nu(E)$ [2506.01640]. 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” [2507.11418]. ## 3. Elliptic-curve murmurations and AI-assisted discovery For elliptic curves over $\mathbb Q$, the central local quantity is the Frobenius trace $$ a_p(E):=p+1-\#E(\mathbb F_p). $$ Fixing a conductor window $I\subset\mathbb R_+$ and a parity $\epsilon\in\{0,1\}$, He, Lee, Oliver, and Pozdnyakov define $$ S(\epsilon,I):=\{\text{isogeny classes }E:N(E)\in I,\ \mathrm{rank}(E)\equiv\epsilon\pmod 2\} $$ and the murmuration $$ m_{\epsilon,I}(p)=\frac1{|S(\epsilon,I)|}\sum_{E\in S(\epsilon,I)}a_p(E). $$ Empirically, as $p$ varies, the graph of $m_{\epsilon,I}(p)$ 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 $I$ of very different sizes [2603.09680]. The same source makes machine-learning interpretability part of the mathematical narrative. Representing each curve by $$ X(E)=(a_{p_1}(E),a_{p_2}(E),\dots,a_{p_n}(E)), $$ PCA produces a first principal vector whose coordinates, plotted against the primes, reproduce the same oscillatory pattern as $m_{\epsilon,I}(p)$. In supervised prediction of rank parity, saliency curves $$ s_p=\left|\partial\,\mathrm{score}(E)/\partial a_p(E)\right| $$ 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 [2603.09680]. The arithmetic interpretation is tied explicitly to Birch–Swinnerton-Dyer and random matrix theory. Since $$ \frac{L'}{L}(E,s)=\sum_p\frac{a_p(E)\log p}{p^s}+\dots, $$ the average of $\sum_{p\le X} a_p(E)/p$ over parity-constrained families heuristically encodes rank information, and differentiating in $X$ recovers the murmuration curve $m_{\epsilon,I}(p)$ [2603.09680]. Under GRH for the family $S(\epsilon,I)$, the same source states a first-order bias formula $$ m_{\epsilon,I}(p)=\frac{c_\epsilon}{\sqrt p}\cos(2\sqrt p+\varphi_\epsilon)+O(p^{-1}), $$ which explains both the $1/\sqrt p$ decay and the oscillatory phase. A separate line of work studies elliptic curves ordered by naive height rather than conductor. For minimal models $$ E:\ y^2=x^3+Ax+B,\qquad H(E)=\max\{4|A|^3,27|B|^2\}, $$ Sawin and Sutherland consider $$ M_X(C_1,C_2)= \mathbb E_{\{E:H(E)\le X\}} \left[ \frac{\log\!\bigl(N(E)\tfrac{C_1+C_2}{2}\bigr)}{N(E)} \sum_{\substack{p\text{ prime}\\ p\in(C_1N(E),\,C_2N(E)]}} \epsilon(E)\,a_p(E) \right]. $$ They conjecture that $M_X(C_1,C_2)$ tends to $\int_{C_1}^{C_2}\rho(u)\,du$, where $\rho(u)$ is given by an explicit double Bessel sum, and they prove a smoothed variant via the Voronoi summation formula [2504.12295]. 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 $E/\mathbb F_q$, Weng’s non-abelian rank-$n$ zeta function is $$ \zeta_{E/\mathbb F_q,n}(s) = \sum_{d\in n\mathbb Z}\sum_V \frac{q^{h^0(X,V)}-1}{|\mathrm{Aut}(V)|}\,q^{-s\deg V}, \qquad \Re(s)>1, $$ with $$ \zeta_{E/\mathbb F_q,n}(s) = \frac{P_{E,n}(T)}{(1-T)(1-q^nT)},\qquad P_{E,n}(T)=1-a_{E/\mathbb F_q,n}T+q^nT^2. $$ Shi and Weng define rank-$n$ averages $f_{r,n}(i)$ by suitable linear normalizations of the coefficients $a_{E/p_i,n}$, and report that the scatter-plots $(i,f_{r,n}(i))$ show exactly the same murmuration patterns, in the same conductor ranges and arithmetic rank, as for $n=1$ [2410.04952]. The theoretical explanation is an asymptotic expansion valid for $n\ge 3$: $$ a_{E/q,n} = (5-n)+(n-1)a_{E/q,1}-(n-1)q+O(q^{-1/2}), \qquad (q\to\infty). $$ After the normalization used in the definition of $f_{r,n}(i)$, this yields $$ \frac{a_{E/p,n}+(n-1)p+n-5}{n-1}\sim a_{E/p,1}, $$ so that, to leading order, $f_{r,n}(i)=f_{r,1}(i)+o(1)$ [2410.04952]. In the same paper, a high-rank Sato–Tate statement is formulated via a renormalized invariant $\Delta_{E/p,n}$ whose limiting law is again the classical $(2/\pi)\sin^2\theta\,d\theta$ 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 $$ \mathbf s_i=\frac{\mathbf v_i}{|\mathbf v_i|}, $$ with maximum-entropy distribution $$ P(\{\mathbf s_i\})= \frac{1}{Z(\{J_{ij}\})} \exp\!\Bigl[\tfrac12\sum_{i,j}J_{ij}\,\mathbf s_i\cdot \mathbf s_j\Bigr] \propto \exp[-E(\{\mathbf s_i\})], $$ equivalent to a classical Heisenberg model with effective Hamiltonian $$ E(\{\mathbf s_i\})=-\tfrac12\sum_{i,j}J_{ij}\,\mathbf s_i\cdot\mathbf s_j. $$ Assuming equal interaction with the $n_c$ nearest neighbors, inference over 21 flocking events gives $n_c\approx 21\pm 2$ and $J\approx 40\text{--}60$, independent of flock size or density; the correlation length $\xi$ is observed to scale linearly with flock size $L$ [1107.0604]. 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 $0$ is the convex hull, shell $1$ consists of birds connected to shell $0$, and so on [1902.08181]. Each bird combines coalignment $$ \mu_i^{\mathrm{coalign}}=\vartheta(\langle \hat v_j\rangle_{j\in B_i}) $$ with a shell-dependent “bounding” term built from shell-mates $$ \mu_i^{\mathrm{bias}}=\langle \hat r_{ij}\rangle_{j\in S_i}, $$ through $$ \mu_i=f_{\kappa(i)}\mu_i^{\mathrm{bias}}+(1-f_{\kappa(i)})\mu_i^{\mathrm{coalign}}, $$ followed by additive directional noise of strength $\phi_n$ [1902.08181]. Calibrated to starling data with $N=1500$ birds and $\phi_n=0.22$, the best-fit parameters are $$ \phi_e=0.883,\qquad a=-0.944,\qquad b=0.056. $$ The resulting density profile falls approximately linearly from $\rho(0)=1$ at the border to $\rho(1)\approx 0.5$ at the center, in agreement with field data showing higher density at the border than near the centre [1902.08181]. 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 $R$, a bird at radius fraction $D=r_p/R$ has anisotropy $$ \Delta I=\frac{I_{\min}}{I_{\max}}=\left(\frac{1-D}{1+D}\right)^3, $$ and simulation indicates an almost linear correspondence $D\leftrightarrow \kappa$, hence a monotonic map $\kappa\leftrightarrow \Delta I$ [1902.08181]. 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 $m$ nearest neighbors in the “toward-COM” half-space, while repulsion acts on up to $m$ neighbors in the opposite half-space [1504.02022]. The continual reshuffling of these topological neighbor sets produces deterministic broadband “internal noise” without any stochastic forcing. Numerically, the velocity-fluctuation correlation length satisfies $\xi\propto L$, the large-$L$ correlation behaves as $C_\infty(r)\sim r^{-\gamma}$ with $\gamma\simeq 0.44$, the power spectral density obeys $S_v(\omega)\sim \omega^{-\alpha_v}$ with $\alpha_v\simeq 1.08$, and for $m=3$ the largest Lyapunov exponent is $\lambda_1\simeq 0.28$ [1504.02022]. 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 $s$: $$ D_t v=(1/\chi)\,s\times v-\nabla P(\rho)-\partial V/\partial v, $$ $$ D_t s=(\mathcal J/v_0^2)\,v\times \nabla^2 v-(\eta/\chi)s, $$ $$ \partial_t\rho=-\nabla\cdot(\rho v). $$ Linearization yields a cubic dispersion relation with two characteristic mode speeds, $$ c_1=\sqrt{\rho_0\sigma},\qquad c_2=\sqrt{\mathcal J/\chi}, $$ corresponding to first sound and second sound [1410.2868]. When $\epsilon=c_2/c_1>\epsilon_c=\sqrt8$, a finite $k$-interval contains only damped modes, implying a “silent” size window $$ 1/k_1