--- title: Murmuration Density Function Analysis url: https://www.emergentmind.com/topics/murmuration-density-function type: topic --- # Murmuration Density Function Analysis The murmuration density function denotes the limiting or family-level profile that governs oscillatory averages of local arithmetic data when the local variable is scaled by the conductor or analytic conductor of the family. In the originating elliptic-curve setting, the core object is the parity-conditioned family average of Frobenius traces, $$ m_{\varepsilon,I}(p)=\frac{1}{|S(\varepsilon,I)|}\sum_{E\in S(\varepsilon,I)} a_p(E), $$ and the murmuration density function is the conjectural scale-invariant limit $$ M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 00}$ is $$ M_{E,\ell,k}=\frac{1}{|E|}\int_E \mathcal D_{\ell,k}(v)\,dv. $$ Under GRH for Dirichlet $L$-functions, the prime-averaged signed Hecke coefficients converge to $M_{E,\ell,k}$ with error $$ O_{E,k,\ell,\varepsilon}\!\big(\ell^{(-\frac15+\varepsilon)a}\big), $$ and the same density arises for the definite quaternion algebra ramified at $\{\infty,\ell\}$. The Maass analogue replaces $\mathcal D_{\ell,k}(v)$ by $$ \mathcal D_{\ell,f}(v) = \frac{\pi}{2f(1)(1-\ell^{-1})} \sum_{r\in\mathbb Z} \Big(\mathbf 1_{\ell\mid r}-\ell^{-1}\Big)\,A_{\ell r}\,\sqrt v\; Q_f\!\left(\frac{r^2}{\ell^2 v}\right), $$ with the corresponding density $M_{E,\ell,f}$ [2606.08353]. The odd-exponent depth-aspect case was computed earlier for $N=\ell^{2a+1}$. There the density is $$ M_{k,\ell}(u) = (-1)^{\frac{k}{2}+1}\frac{2\pi}{(k-1)(\ell-1)} \sum_{\substack{t\in\mathbb Z\\ |t|<2\ell\sqrt u}} \Big(\mathbf 1_{\ell\mid t}-\ell^{-1}\Big) \left(\prod_{\substack{p\neq 2\\ p\nmid \ell t}} \frac{p^2-p-1}{p(p-1)}\right) \sqrt{u-\frac{t^2}{4\ell^2}}\; U_{k-2}\!\left(\frac{t}{2\ell\sqrt u}\right), $$ and one has $$ \mathcal M_{k,\ell,E}(a) = \frac{1}{|E|}\int_E M_{k,\ell}(u)\,du + O_{E,k,\ell,C}(a^{-C}) $$ for any fixed $C>0$, with a GRH improvement to $O_{E,k,\ell,\varepsilon}((\ell^{2a+1})^{-1/2+\varepsilon})$ [2603.25564]. The later even-exponent paper shows that the density $M_{E,\ell,k}$ agrees with the density previously obtained for odd conductor exponents, yielding a unified depth-aspect density for cusp forms of conductor $\ell^n$ as $n\to\infty$ [2606.08353]. In the weight aspect for level-$1$ holomorphic forms, a different explicit density appears. Averaging the signed coefficients $\varepsilon_f\lambda_f(p)$ over weights $k\asymp K$ and primes with $p/K^2\in E=[A,B]$, the normalized average satisfies $$ \mathcal C(K;E,h) = \frac{1}{|E|}\int_E M_K(y)\,dy+o\!\left(\frac1K\right), \qquad M_K(y)=\frac{1}{2K\sqrt y}, $$ so that $$ \mathcal C(K;E,h) = \frac{1}{K}\cdot\frac{\sqrt B-\sqrt A}{|E|} + o\!\left(\frac1K\right). $$ This formulation isolates a universal $y^{-1/2}$ density in the rescaled prime variable $y=p/K^2$ [2507.11418]. A further explicit family is the square-free level aspect for weight-$2$ newforms. There the same functional form governs both prime-indexed coefficients and square-of-prime coefficients: $$ M(\xi) = A\sqrt{\xi} + B\sum_{1\le r\le 2\sqrt{\xi}} C(r)\sqrt{4\xi-r^2} - \pi\xi. $$ The only change is the scale variable, namely $\xi=P/X$ in the prime case and $\xi=P^2/X$ in the $p^2$ case. The paper states that the shape of the murmuration density is the same in both situations [2507.00738]. ## 3. Dirichlet characters and Hecke $L$-functions For Dirichlet characters, the term “murmuration density” acquires both oscillatory and distributional meanings. In the complex-character family with prime conductor $N$, the limiting densities in the conductor-ratio variable $x$ are $$ M_+(x;y)=\cos(2\pi y/x), \qquad M_-(x;y)=-i\sin(2\pi y/x), $$ and the corresponding murmuration functions over a geometric window are $$ \mathcal M_\pm(y;c)=\int_1^c M_\pm(x;y)\,dx. $$ For short windows in $N$, the limits collapse to $\cos(2\pi y)$ and $-i\sin(2\pi y)$. In the real quadratic family, the density is a distribution $$ M(u) = \frac{\sqrt 2}{2} \sum_{\substack{a\ge 1\\ (a,2)=1}} \frac{\mu(a)}{a^2} \sum_{m=1}^\infty (-1)^m \cos\!\left(\frac{\pi m^2}{a^2u}-\frac{\pi}{4}\right), $$ and for every smooth compactly supported $\Phi$ one has $$ M_\Phi(y,\delta)=\int \Phi(t)M(y/t)\,dt. $$ The same paper proves $$ \lim_{y\to 0^+} M_\Phi(y,\delta)=0, \qquad \lim_{y\to \infty} M_\Phi(y,\delta)= -\frac{2}{\pi^2}\widetilde\Phi(0), $$ which it interprets as interpolation of the phase transition in the $1$-level density for a symplectic family [2307.00256]. For Hecke $L$-functions of imaginary quadratic fields associated to non-trivial class-group characters, the prime-aspect average admits a pointwise almost-periodic formula before prime-averaging: $$ G(p,X,Y) = c(p)\sum_{1\le y<2\sqrt{\xi}} \delta_y(p)\,M_y(\xi) + M_-(\xi) + O_{\delta_\xi}\!\left(X^{-\delta_0+\varepsilon}\right), \qquad \xi=p/X. $$ After averaging over primes in a short interval, one obtains a genuine density $$ M(\Xi) = \overline c\sum_{1\le y<2\sqrt{\Xi}} \overline M_y(\Xi) + M_-(\Xi), $$ together with a universal Bessel expansion $$ M(\Xi) = \frac{11\pi\,\zeta(2)\,\overline c}{16A}\,\sqrt{\Xi} \sum_{d=1}^\infty \frac{Q(d)}{d}\sum_{m=1}^\infty J_0\!\left(\frac{4\pi m\sqrt{\Xi}}{d}\right) -\frac12. $$ For the associated $\Phi$-smoothed murmuration function, the paper proves $$ M_\Phi(0)=0, \qquad \lim_{\Xi\to\infty} M_\Phi(\Xi)=-\frac12. $$ A distinctive feature of this family is the pronounced almost periodic dependence on the prime variable $p$, which the paper describes as allowing the murmuration to be described without averaging over primes [2503.17967]. These examples show that the MDF can be a literal integrand, a distribution tested against smooth weights, or an averaged reduction of an almost-periodic prime-level signal. What remains common is the rescaled local variable and a family average that stabilizes to a structured profile. ## 4. Elliptic-curve variants: empirical densities, BSD stratification, and height ordering In a second elliptic-curve line of work, the murmuration density function is defined empirically rather than by a closed analytic formula. For a conductor-windowed family $\mathcal F$ and normalized traces $$ x_p(E)=\frac{a_p(E)}{2\sqrt p}, $$ the per-prime empirical distribution is $$ f_{p,\mathcal F}(x)=\frac{1}{|\mathcal F|}\sum_{E\in\mathcal F}\delta(x-x_p(E)), $$ and its first moment is the normalized murmuration profile $$ M_x(p;\mathcal F)=\int x\,f_{p,\mathcal F}(x)\,dx =\frac{1}{|\mathcal F|}\sum_{E\in\mathcal F}x_p(E). $$ In this formulation, the “density” is the empirical measure $f_{p,\mathcal F}$, while the usual murmuration curve is its mean as a function of $p$ [2603.04604]. That paper studies BSD invariants in sliding conductor windows. It reports three results: the BSD invariants themselves do not exhibit murmuration-type oscillations when averaged in sliding conductor windows; within a fixed rank, stratification by Tamagawa product, analytic order of the Tate–Shafarevich group, or real period yields significantly different murmuration profiles, with $p$-values less than $0.001$ against permutation null models; and the Tate–Shafarevich modulation persists after simultaneously controlling for $L(E,1)$, the real period, and the conductor [2603.04604]. The modulation is described as a pure mean shift: for the $\#\Sha$ stratification, the paper finds identical variance, skewness, and kurtosis across strata, while the difference in means is concentrated at small primes and changes sign once. A different elliptic-curve variant orders curves by naive height rather than conductor. For the family $$ F(X)=\{E: H(E)\le X\}, $$ the conjectural prime-restricted murmuration density is an explicit Bessel series, $$ M(u) = 2\pi \sqrt u \sum_{\substack{q\ge 1\\ q\ \mathrm{squarefree}}} \sum_{m\ge 1} \frac{\mu(\gcd(m,q))}{q\,m\,\varphi(q/\gcd(m,q))} J_1\!\left(\frac{4\pi\sqrt u\,m}{q}\right) \prod_{p\mid q}\check\ell_{p,2v_p(m)} \prod_{\substack{p\mid m\\ p\nmid q}} \ell_{p,2v_p(m)}. $$ The paper proves the corresponding statement for a smooth $n$-sum restricted to integers with no small prime factors and states that the prime-restricted sharp-cutoff formula is conjectural. It describes this as the first work to give an explicit formula for the murmuration density of a family of elliptic curves, in any ordering [2504.12295]. These two directions exhibit two distinct senses of MDF within elliptic-curve arithmetic. One is empirical and distributional, with the density realized by $f_{p,\mathcal F}(x)$ and studied through moments and permutation tests. The other is an explicit Bessel-kernel formula with $p$-adic local factors, motivated by Voronoi summation and proved in a rough-integer model. ## 5. Function-field murmurations and exact finite-sum densities Over function fields, the MDF becomes completely explicit in a different manner. For the family $$ E_D:\ y^2=x^3+x+D(t), $$ with $D\in\mathbb F_q[t]$ monic squarefree of degree $5$, the global $L$-function is a polynomial whose reciprocal roots $\alpha_j$ all satisfy $|\alpha_j|=q$. Writing $\zeta_j=\alpha_j/q$, the unitarized polynomial $L_{\mathrm{unit}}(z)$ has integer coefficients and all roots on the unit circle; by Kronecker’s theorem it factors into cyclotomic polynomials. Each curve therefore has a cyclotomic $L$-polynomial type $\lambda$ [2603.13802]. For good places $v$ of degree $d$, the Frobenius trace is $$ a_v(E_D)=-\sum_j \alpha_j^d, $$ so the unitarized trace is $-\sum_j \zeta_j^d$. If $p_d(\lambda)$ denotes the degree-$d$ power sum of the unitarized roots of type $\lambda$, and if $f_{\lambda,s}$ is the fraction of curves of type $\lambda$ in the stratum $|\Sha|=s$, then the paper proves the exact reweighting identity $$ M_s(d,q)=-\sum_\lambda f_{\lambda,s}\,p_d(\lambda). $$ No asymptotic limit is involved: this is an exact finite sum [2603.13802]. In this family, BSD is a theorem and simplifies drastically. The Mordell–Weil rank is $0$ in rank-$0$ cases, the regulator is $1$, global torsion is trivial, and all Tamagawa numbers are $1$, so $$ |\Sha(E_D)|=L(E_D,1/q). $$ Because $L(E_D,1/q)=L_{\mathrm{unit}}(1)$ depends only on the cyclotomic type, the paper concludes that the $|\Sha|$ modulation of murmurations is entirely a composition effect: different $|\Sha|$ strata have different mixtures of cyclotomic types, and therefore different averages of the power sums $p_d(\lambda)$ [2603.13802]. The paper also states that within each $|\Sha|$ stratum there are joint cells, meaning distinct $L$-polynomial types with the same $|\Sha|$ but different trace profiles. Accordingly, the murmuration profile carries arithmetic information strictly finer than $|\Sha|$ alone. This is a particularly sharp formulation of MDF: the density is an exact combinatorial average of cyclotomic power sums, and the oscillatory behavior is visible in the place degree $d$ rather than in a scaled prime variable. ## 6. Methods, interpretation, and unresolved issues The literature presents several distinct mechanisms for producing murmuration densities. In the original elliptic-curve work, the signal is extracted from large arithmetic datasets and then analyzed with PCA, saliency curves, and convolutional filters, with averaging playing a decisive role in separating the murmuration from dominant rank-correlated features such as Mestre–Nagao sums [2603.09680]. In modular and Maass settings, the MDF is derived from trace formulas: the Eichler–Selberg or Yamauchi–Skoruppa–Zagier formula in the holomorphic depth aspect, the simple trace formula for the definite quaternion algebra, the adelic Arthur–Selberg trace formula for Maass forms, and the Petersson trace formula in the weight aspect [2606.08353]. In the height-ordered elliptic-curve problem, the Bessel kernel arises from the GL$(2)$ Voronoi summation formula [2504.12295]. Several recurring clarifications are important. First, the MDF is not the same object as the Sato–Tate distribution for a single curve. The original elliptic-curve paper explicitly states that it does not derive a formula connecting $m_{\varepsilon,I}(p)$ to Sato–Tate or Chebotarev densities, and it distinguishes the family-average signal from usual statements about the distribution of $a_p(E)/(2\sqrt p)$ for an individual curve [2603.09680]. Second, the MDF is not exhausted by simple rank bias. The same paper emphasizes that, although higher-rank curves tend to have negative aggregate discrepancies individually, the family averages $m_{\varepsilon,I}(p)$ oscillate around zero with a scale-invariant profile, which it states “defies the expectation” that rank bias alone would dominate family averages. The BSD-stratification study sharpens this point by exhibiting mean-shift modulation at fixed rank and under multiple controls [2603.04604]. Third, the status of the MDF varies by family: it is conjectural as a scale-invariant limit in the original conductor-ordered elliptic-curve setting, explicit and proved in several automorphic families, empirical in some statistical studies, and exact in the function-field cyclotomic model [2603.09680]. The principal open problems are equally consistent across the literature. The originating elliptic-curve paper calls for a rigorous definition and proof of the existence of $M_\varepsilon(u)$ with quantified asymptotics and error terms, together with an identification of the arithmetic mechanism—trace formulas, spectral decompositions, or explicit formulas—behind the observed oscillations and scale invariance [2603.09680]. The depth-aspect work formulates precise GRH-dependent asymptotics but still points toward deeper random-matrix and low-lying-zero interpretations, including possible relations to one-level density phenomena [2606.08353]. The height-ordered elliptic-curve paper leaves the prime-restricted sharp-cutoff density conjectural, even though the smooth rough-integer variant is proved [2504.12295]. Taken together, these works show that “murmuration density function” is not a single formula but a family of closely related objects. In each case it is the structured limit, exact formula, or empirical distribution governing local coefficients after conductor-scale rescaling and family averaging. This suggests that the MDF is best understood as a family-level statistic at the intersection of arithmetic statistics, trace formulas, explicit formulas, and low-lying-zero phenomena. Source: https://www.emergentmind.com/topics/murmuration-density-function