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arxiv: 2606.23681 · v1 · pith:ACX3AECYnew · submitted 2026-06-22 · 🧮 math.NT

Rank Amplification for Shifted Equal Values of Euler's Totient Function

Eric Li (Trinity College , University of Cambridge) This is my paper

Pith reviewed 2026-06-26 06:59 UTC · model grok-4.3

classification 🧮 math.NT MSC 11A2511N25
keywords Euler totient functionshifted equal valuessame-support familysmooth-totient theoremdivisor convolutionsupplier systemsfriable tuples
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The pith

The number of n ≤ x with φ(n) = φ(n+1) is ≪ x exp{-(1/2-o(1)) √(log x log_2 x)}.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an upper bound on the number of integers n up to x where Euler's totient function takes the same value at n and at n plus a small shift h. For the unit shift h=1 it obtains a saving of exp{-(1/2-o(1)) times the square root of log x times log log x} over the trivial bound of x. The result is obtained by decomposing S_h^φ(x) into a contribution from the Graham-Holt-Pomerance same-support family above a cutoff and a smaller error term. A sympathetic reader would care because the bound shows that totient collisions occur much less often than the total number of integers up to x, even for nearby arguments.

Core claim

For every fixed integer J ≥ 1, uniformly for 1 ≤ h ≤ exp(G/√J) with G = √(log x, A), A = log_3 x + log_4 x - log 2, V = log x / G and Y_J = exp(√J G), one has S_h^φ(x) = D_{h,>Y_J}^φ(x) + O_J(x exp{-√J G + o_J(V)}), where D is the above-cutoff part of the classical same-support family; this is empty for odd h. A moving choice J ≍ log_2 x / log_3 x recovers the unit-shift bound.

What carries the argument

Labelled supplier systems for the shifted divisor convolution that permit an injective encoding of large supplier products into weighted friable tuples, used in tandem with the smooth-totient theorem.

If this is right

  • The error term is o(x) uniformly in the stated range of h.
  • When h is odd the same-support term D vanishes, so the full count equals the error term.
  • Larger fixed J tightens the error at the expense of a shorter admissible interval for h.
  • A slowly growing J yields both the unit-shift saving and a decomposition valid for a wider but still sub-exponential range of shifts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may carry over to other arithmetic functions whose level sets admit analogous convolution decompositions.
  • Stronger results on the smoothness of totient values would immediately enlarge the admissible range of h or improve the saving.
  • The same-support contribution is expected to remain the dominant term even for moderately larger h, though this lies outside the paper's uniform range.

Load-bearing premise

Labelled supplier systems for the shifted divisor convolution exist and satisfy the stated injective encoding property into weighted friable tuples.

What would settle it

An explicit count or construction showing that the number of n ≤ x with φ(n) = φ(n+1) exceeds x exp{-(1/2 - ε) √(log x log_2 x)} for some fixed ε > 0 and a sequence of x tending to infinity would falsify the unit-shift claim.

read the original abstract

Let $S_h^\varphi(x)$ denote the number of integers $n\le x$ for which $\varphi(n)=\varphi(n+h)$. For the unit shift, we prove $S_1^\varphi(x)\ll x\exp{-(1/2-o(1))\sqrt{\log x,\log_2 x}}$. More generally, put $A=\log_3 x+\log_4 x-\log 2$, $G=\sqrt{\log x,A}$, and $V=\log x/G$. For every fixed integer $J\ge 1$, uniformly for $1\le h\le \exp{G/\sqrt{J}}$, we obtain $S_h^\varphi(x)=D_{h,>Y_J}^\varphi(x)+O_J(x\exp{-\sqrt{J},G+o_J(V)})$, where $Y_J=\exp{\sqrt{J},G}$. Here $D_{h,>Y_J}^\varphi(x)$ is the above-cutoff part of the classical Graham--Holt--Pomerance same-support family; it is empty for odd $h$. A moving choice $J\asymp \log_2 x/\log_3 x$ gives the unit-shift estimate and an analogous decomposition for a uniform range of shifts. The proof combines the smooth-totient theorem of Banks--Friedlander--Pomerance--Shparlinski with labelled supplier systems, a shifted divisor convolution, and an injective encoding of large supplier products into weighted friable tuples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript claims to prove that S_h^φ(x), the number of n ≤ x with φ(n) = φ(n + h), equals the above-cutoff contribution D_{h,>Y_J}^φ(x) from the Graham–Holt–Pomerance same-support family plus an error O_J(x exp{−√J G + o_J(V)}), uniformly for 1 ≤ h ≤ exp(G/√J) with the indicated parameters G, V, Y_J. For the unit shift h = 1 a moving choice J ≍ log₂x / log₃x recovers the bound S_1^φ(x) ≪ x exp{−(1/2 − o(1)) √(log x, log₂ x)}. The argument combines the smooth-totient theorem of Banks–Friedlander–Pomerance–Shparlinski with a new construction of labelled supplier systems that encode large supplier products injectively into weighted friable tuples for the shifted divisor convolution.

Significance. If the new constructions are valid, the result would give a quantitatively stronger upper bound on the number of solutions to φ(n) = φ(n + h) for small h than previous work, by isolating the main contribution from the classical same-support family and controlling the remainder via the smooth-totient theorem. The explicit error term and the uniform range in h are potentially useful for further applications in the distribution of totient values.

major comments (1)
  1. [Abstract, final paragraph] Abstract, final paragraph: the existence, density estimates, and injectivity of the labelled supplier systems for the shifted divisor convolution (asserted to hold for 1 ≤ h ≤ exp(G/√J)) are load-bearing for the error term O_J(x exp{−√J G + o_J(V)}); the manuscript must supply either a self-contained construction or a reference establishing these properties, as they are not standard objects and the claimed saving exp{−(1/2 − o(1)) √(log x, log₂ x)} does not follow without them.
minor comments (2)
  1. The notation √(log x, log₂ x) and exp{−√J, G + o_J(V)} should be clarified (e.g., whether the comma denotes multiplication) for readability.
  2. The definition of the parameters A, G, V, Y_J and the precise statement of the Graham–Holt–Pomerance family D_{h,>Y_J}^φ(x) would benefit from an explicit reference or short recap in the introduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comment on our manuscript. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract, final paragraph] Abstract, final paragraph: the existence, density estimates, and injectivity of the labelled supplier systems for the shifted divisor convolution (asserted to hold for 1 ≤ h ≤ exp(G/√J)) are load-bearing for the error term O_J(x exp{−√J G + o_J(V)}); the manuscript must supply either a self-contained construction or a reference establishing these properties, as they are not standard objects and the claimed saving exp{−(1/2 − o(1)) √(log x, log₂ x)} does not follow without them.

    Authors: The labelled supplier systems are a new construction introduced in this manuscript. Their existence, density estimates, and injectivity for the shifted divisor convolution, uniformly in the range 1 ≤ h ≤ exp(G/√J), are established self-containedly in Sections 3--5 via the injective encoding of large supplier products into weighted friable tuples, combined with the Banks--Friedlander--Pomerance--Shparlinski smooth-totient theorem. We will revise the final paragraph of the abstract to include an explicit reference to these sections. revision: yes

Circularity Check

0 steps flagged

No circularity: bound derived from external theorem plus internal constructions

full rationale

The derivation decomposes S_h^φ(x) via the external smooth-totient theorem of Banks–Friedlander–Pomerance–Shparlinski, then controls the tail using labelled supplier systems and an injective encoding into friable tuples. These objects are introduced and asserted to exist in the present work rather than imported via self-citation; the resulting error term O_J(x exp{−√J,G + o_J(V)}) is not obtained by fitting a parameter to the target quantity or by renaming a known result. No equation reduces to its own input by definition, and the central estimate for the unit shift follows from the decomposition plus a moving choice of J without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Based solely on the abstract; the proof rests on one external theorem and introduces two new combinatorial objects whose properties are asserted but not derived here.

axioms (1)
  • domain assumption smooth-totient theorem of Banks--Friedlander--Pomerance--Shparlinski
    Invoked as a black-box input for controlling smooth values of φ.
invented entities (2)
  • labelled supplier systems no independent evidence
    purpose: Encode large supplier products into weighted friable tuples for the shifted convolution
    New tool introduced in the abstract to obtain the error term after the cutoff Y_J.
  • weighted friable tuples no independent evidence
    purpose: Represent the injective encoding of supplier products
    New encoding device mentioned in the abstract.

pith-pipeline@v0.9.1-grok · 5801 in / 1403 out tokens · 33668 ms · 2026-06-26T06:59:29.647920+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

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