Chowla–Elliott correlations via divisibility graphs

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Chowla–Elliott correlations via divisibility graphs

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Recursos

Take the ordinary average of λ(n)λ(n+h) up to X and follow the four moves that bound it by a power of log X. At each step you should be able to say what it buys and why weight one makes it harder than the logarithmic version.

Why ordinary averages are harder [0:20]

A logarithmic average mixes many multiplicative scales together. An ordinary average has to control the terminal scale, at every cutoff X. Dilation shows where this hurts [0:50]: λ is completely multiplicative and λ(d)² = 1, so the pair at dm, d(m+h) has the same product as the pair at m, m+h. That gives the divisibility graph, with an edge n → n+hd whenever d | n. But substituting n = dm shrinks the cutoff from T to T/d [1:11]. To fix this, the proof sorts divisors into bins, so that every T/(qd) lands in (Xe^{−η}, X] and F(X) comes back up to O(η).

The four moves

Dilate along divisors, as above.
Center [1:37]: replace 1_{p|n} by 1_{p|n} − 1/p, which has mean zero for a uniform residue mod p. When you expand the product, the full-divisibility term returns S₀F(X). The short-interval estimate of Matomäki–Radziwiłł–Tao controls the rest. The proof works with independent residues per prime and compares them with integer averages through bounded independence (Braverman).
Trace [3:10]: a high trace is a sum over closed walks. A prime that appears on only one edge makes the walk's expectation zero (think-first at [2:37]), so every surviving prime appears at least twice. Repeated primes impose congruences, each saving about (1+L)/H₀. Prohibited paths are deleted, and a forest code counts what remains.
Transfer [3:47]: ‖M‖^{2m} ≤ Tr((M*M)^m), so a bound N·R^{2m} gives ‖M‖ ≤ N^{1/(2m)}R, and the dimension N fades as m grows. A transfer lemma [4:20] moves the bound from the nonbacktracking operator to the projected edge sum: at most 3·max{a, b, ρ}. Testing against √w·λ gives the log saving.

The Elliott version [4:45]

For a general function f, nonpretentiousness gives divergence but no rate, so the order of limits flips. Fix a scale B first, with a weighted core band and a center band centered by θ/p. A bias forces a raw sum of size c·L₀/B, but the graph estimate gives L₀·B^{−1−c*/2}. Let X → ∞ with B fixed, then let B grow, and the bias cannot survive. Next, read Sections 13–17 of the paper to see how the first-visit tree keeps the center primes' cancellation alive [5:29].

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Converse theorem for elliptic curves with 2-torsion

Hilbert's tenth problem over the rationals

Why Catalan's constant G is irrational

Goldfeld's conjecture and the 2∞-Selmer converse

Average rank of elliptic curve twists

Chowla–Elliott correlations via divisibility graphs

The Grothendieck–Teichmüller Lie algebra and the Deligne–Drinfeld conjecture

Milnor K-theory and field reconstruction mod p

Milnor K-theory and birational anabelian reconstruction

Bogomolov–Pop reconstruction: the uniform bound along a pencil

Fontaine–Mazur modularity at the prime 2

The dimension of the two-adic Hecke algebra

Pro-modularity at 2 without de Rham conditions

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Poisson-Dirichlet law for prime factors of p−1

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Independence of largest prime factors in consecutive integers

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Hecke eigensheaves with tame ramification

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Global Arthur parameters via cuspidal excursion data

Nilpotent orbit decomposition of cuspidal automorphic functions

Rationality of the canonical Arthur filtration

Hecke eigensheaves with ramification at marked points

Temperedness at ramified places for exceptional groups

Restricted geometric Langlands in positive characteristic

Equidistribution of prime-degree torus packets

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Equidistribution of quartic torus packets as orders grow

Zilber–Pink for curves in A2: elliptic squares

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Log abundance for compact Kähler spaces

Log abundance for log canonical pairs in characteristic zero

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