Murmuration Density Function Analysis

• Murmuration density function is a scale-invariant profile that represents the limiting family average of local arithmetic coefficients after rescaling by the analytic conductor.
• It is derived both empirically and via trace formulas across elliptic curves, modular forms, and function-field models, revealing oscillatory behaviors and explicit analytic densities.
• The function uncovers subtle stratification effects—including rank bias and BSD invariants—and motivates rigorous study of asymptotic behavior and trace analytic mechanisms.

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

$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$

with $u=p/X$ (He et al., 10 Mar 2026). Subsequent works make the notion explicit in several automorphic and arithmetic families, where the density is given either by closed formulas, by exact finite sums, or by empirical per-prime distributions (Tomczak, 6 Jun 2026).

1. Foundational definition in the elliptic-curve setting

For an elliptic curve $E$ and a prime $p$, the Frobenius trace is

$a_p(E):=p+1-\#E_p,$

where $\#E_p$ is the number of solutions of $E$ modulo $p$ plus the point at infinity. The Hasse bound implies $|a_p(E)|\le 2\sqrt p$. The original murmuration dataset is organized by isogeny classes and rank parity. Writing $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$0 for the conductor, fixing an interval $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$1 and a parity $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$2, one sets

$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$3

The family average

$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$4

is the basic murmuration signal (He et al., 10 Mar 2026).

The defining empirical feature is scale invariance. When $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$5 is plotted for disjoint conductor windows of comparable relative width and the $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$6-axis is rescaled to a common length, the prominent oscillations—especially peaks and zero-crossings—occur at the same scaled locations. This motivates the scale-invariant profile

$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$7

In the numerical experiments, conductor windows include $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$8, $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$9, $u=p/X$0, and $u=p/X$1, and the prime range drawn in each figure is $u=p/X$2 (He et al., 10 Mar 2026).

Several normalization choices are built into this definition. Isogeny classes are weighted uniformly by $u=p/X$3. No smoothing kernel is applied in the basic signal; the paper emphasizes the raw average $u=p/X$4 with the prime axis rescaled. A smoothed variant can be written down, but it is not used in that paper. The observed phenomenon is therefore not a smoothing artifact but a property of the unsmoothed family average (He et al., 10 Mar 2026).

The original study also links the murmuration profile to machine-learning representations. Curves are encoded by

$u=p/X$5

with $u=p/X$6 for primes $u=p/X$7 in one PCA example, and the first principal direction has entries exhibiting the same oscillatory structure as $u=p/X$8. Saliency curves and convolutional filters also detect related signals, although their outputs are often dominated by classical Mestre–Nagao sums, which can obscure the subtler murmuration profile. In this sense, the murmuration density function emerged from averaging and interpretability rather than from an a priori analytic ansatz (He et al., 10 Mar 2026).

2. Explicit analytic densities in automorphic families

After the initial elliptic-curve discovery, several papers computed explicit murmuration densities in automorphic families. In these settings, the MDF is no longer merely an empirical profile but the main term in a prime-averaged asymptotic formula.

Family	Scale variable	Density object
Elliptic curves by conductor window and parity	$u=p/X$9	$E$0
Holomorphic newforms of conductor $E$1	$E$2	$E$3, $E$4
Maass newforms of conductor $E$5	$E$6	$E$7, $E$8
Weight-aspect level-$E$9 forms	$p$0	$p$1
Square-free level newforms, $p$2 or $p$3 coefficients	$p$4 or $p$5	$p$6

For holomorphic cusp forms of conductor $p$7 and fixed even weight $p$8, the depth-aspect density is

$p$9

and the averaged density over a compact interval $a_p(E):=p+1-\#E_p,$0 is

$a_p(E):=p+1-\#E_p,$1

Under GRH for Dirichlet $a_p(E):=p+1-\#E_p,$2-functions, the prime-averaged signed Hecke coefficients converge to $a_p(E):=p+1-\#E_p,$3 with error

$a_p(E):=p+1-\#E_p,$4

and the same density arises for the definite quaternion algebra ramified at $a_p(E):=p+1-\#E_p,$5. The Maass analogue replaces $a_p(E):=p+1-\#E_p,$6 by

$a_p(E):=p+1-\#E_p,$7

with the corresponding density $a_p(E):=p+1-\#E_p,$8 (Tomczak, 6 Jun 2026).

The odd-exponent depth-aspect case was computed earlier for $a_p(E):=p+1-\#E_p,$9. There the density is

$\#E_p$0

and one has

$\#E_p$1

for any fixed $\#E_p$2, with a GRH improvement to $\#E_p$3 (Burrin et al., 26 Mar 2026). The later even-exponent paper shows that the density $\#E_p$4 agrees with the density previously obtained for odd conductor exponents, yielding a unified depth-aspect density for cusp forms of conductor $\#E_p$5 as $\#E_p$6 (Tomczak, 6 Jun 2026).

In the weight aspect for level-$\#E_p$7 holomorphic forms, a different explicit density appears. Averaging the signed coefficients $\#E_p$8 over weights $\#E_p$9 and primes with $E$0, the normalized average satisfies

$E$1

so that

$E$2

This formulation isolates a universal $E$3 density in the rescaled prime variable $E$4 (Kuan et al., 15 Jul 2025).

A further explicit family is the square-free level aspect for weight-$E$5 newforms. There the same functional form governs both prime-indexed coefficients and square-of-prime coefficients:

$E$6

The only change is the scale variable, namely $E$7 in the prime case and $E$8 in the $E$9 case. The paper states that the shape of the murmuration density is the same in both situations (Kundu et al., 1 Jul 2025).

3. Dirichlet characters and Hecke $p$0-functions

For Dirichlet characters, the term “murmuration density” acquires both oscillatory and distributional meanings. In the complex-character family with prime conductor $p$1, the limiting densities in the conductor-ratio variable $p$2 are

$p$3

and the corresponding murmuration functions over a geometric window are

$p$4

For short windows in $p$5, the limits collapse to $p$6 and $p$7. In the real quadratic family, the density is a distribution

$p$8

and for every smooth compactly supported $p$9 one has

$|a_p(E)|\le 2\sqrt p$0

The same paper proves

$|a_p(E)|\le 2\sqrt p$1

which it interprets as interpolation of the phase transition in the $|a_p(E)|\le 2\sqrt p$2-level density for a symplectic family (Lee et al., 2023).

For Hecke $|a_p(E)|\le 2\sqrt p$3-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:

$|a_p(E)|\le 2\sqrt p$4

After averaging over primes in a short interval, one obtains a genuine density

$|a_p(E)|\le 2\sqrt p$5

together with a universal Bessel expansion

$|a_p(E)|\le 2\sqrt p$6

For the associated $|a_p(E)|\le 2\sqrt p$7-smoothed murmuration function, the paper proves

$|a_p(E)|\le 2\sqrt p$8

A distinctive feature of this family is the pronounced almost periodic dependence on the prime variable $|a_p(E)|\le 2\sqrt p$9, which the paper describes as allowing the murmuration to be described without averaging over primes (Wang, 23 Mar 2025).

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 $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$00 and normalized traces

$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$01

the per-prime empirical distribution is

$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$02

and its first moment is the normalized murmuration profile

$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$03

In this formulation, the “density” is the empirical measure $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$04, while the usual murmuration curve is its mean as a function of $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$05 (Wachs, 4 Mar 2026).

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 $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$06-values less than $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$07 against permutation null models; and the Tate–Shafarevich modulation persists after simultaneously controlling for $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$08, the real period, and the conductor (Wachs, 4 Mar 2026). The modulation is described as a pure mean shift: for the $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$09 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

$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$10

the conjectural prime-restricted murmuration density is an explicit Bessel series,

$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$11

The paper proves the corresponding statement for a smooth $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$12-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 (Sawin et al., 16 Apr 2025).

These two directions exhibit two distinct senses of MDF within elliptic-curve arithmetic. One is empirical and distributional, with the density realized by $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$13 and studied through moments and permutation tests. The other is an explicit Bessel-kernel formula with $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$14-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

$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$15

with $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$16 monic squarefree of degree $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$17, the global $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$18-function is a polynomial whose reciprocal roots $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$19 all satisfy $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$20. Writing $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$21, the unitarized polynomial $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$22 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 $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$23-polynomial type $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$24 (Wachs, 14 Mar 2026).

For good places $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$25 of degree $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$26, the Frobenius trace is

$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$27

so the unitarized trace is $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$28. If $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$29 denotes the degree-$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$30 power sum of the unitarized roots of type $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$31, and if $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$32 is the fraction of curves of type $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$33 in the stratum $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$34, then the paper proves the exact reweighting identity

$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$35

No asymptotic limit is involved: this is an exact finite sum (Wachs, 14 Mar 2026).

In this family, BSD is a theorem and simplifies drastically. The Mordell–Weil rank is $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$36 in rank-$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$37 cases, the regulator is $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$38, global torsion is trivial, and all Tamagawa numbers are $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$39, so

$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$40

Because $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$41 depends only on the cyclotomic type, the paper concludes that the $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$42 modulation of murmurations is entirely a composition effect: different $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$43 strata have different mixtures of cyclotomic types, and therefore different averages of the power sums $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$44 (Wachs, 14 Mar 2026).

The paper also states that within each $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$45 stratum there are joint cells, meaning distinct $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$46-polynomial types with the same $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$47 but different trace profiles. Accordingly, the murmuration profile carries arithmetic information strictly finer than $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$48 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 $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$49 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 (He et al., 10 Mar 2026). 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 (Tomczak, 6 Jun 2026). In the height-ordered elliptic-curve problem, the Bessel kernel arises from the GL$$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$50 Voronoi summation formula (Sawin et al., 16 Apr 2025).

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 $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$51 to Sato–Tate or Chebotarev densities, and it distinguishes the family-average signal from usual statements about the distribution of $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$52 for an individual curve (He et al., 10 Mar 2026). 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 $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$53 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 (Wachs, 4 Mar 2026). 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 (He et al., 10 Mar 2026).

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 $$M_\varepsilon(u)=\lim_{X\to\infty} m_{\varepsilon,[X,2X]}(uX),\qquad 0<u\le 1,$$54 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 (He et al., 10 Mar 2026). 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 (Tomczak, 6 Jun 2026). The height-ordered elliptic-curve paper leaves the prime-restricted sharp-cutoff density conjectural, even though the smooth rough-integer variant is proved (Sawin et al., 16 Apr 2025).

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.