Number of non-primes in the set of units modulo $n$

Abhijit A J; A. Satyanarayana Reddy

arxiv: 1907.09908 · v1 · pith:RSOV3UGKnew · submitted 2019-07-07 · 🧮 math.GM

Number of non-primes in the set of units modulo n

Abhijit A J , A. Satyanarayana Reddy This is my paper

Pith reviewed 2026-05-25 01:08 UTC · model grok-4.3

classification 🧮 math.GM

keywords arithmetic functionsEuler totient functionunits modulo nnon-prime integerscoprime numbersnumber theory

0 comments

The pith

The paper defines tilde-varphi(n) as the count of non-prime integers up to n that remain coprime to n and examines its properties.

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

Authors introduce the function tilde-varphi(n) that counts integers m from 1 to n with gcd(m,n)=1 yet m not prime. This refines Euler's totient by separating the prime and non-prime elements within the group of units modulo n. The work derives several arithmetic properties of the new function from its definition. A sympathetic reader would care because the distinction isolates the contribution of composite residues, which may clarify counting problems that mix primes and composites in modular settings.

Core claim

The arithmetic function tilde-varphi(n) is defined to equal the number of natural numbers m satisfying 1 ≤ m ≤ n, (m,n)=1 and m not prime; the manuscript studies the various properties this function possesses.

What carries the argument

The function tilde-varphi(n), defined as the cardinality of non-prime elements among the units modulo n.

Load-bearing premise

Defining and studying the count of numbers that are both coprime to n and non-prime produces results that go beyond routine extensions of Euler's totient function.

What would settle it

A direct enumeration for a small composite n such as 15 that yields a value for tilde-varphi(15) inconsistent with any formula or relation claimed for the function.

read the original abstract

In this work, we studied various properties of arithmetic function $\tilde{\varphi}$, where $\tilde{\varphi}(n)=|\{m\in \mathbb{N} | 1\le m\le n, (m,n)=1, \mbox{$m$ is not a prime}\}|.$

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.

Desk Editor's Note private letter to a colleague

This paper defines a minor variant of Euler's totient that excludes primes from the units modulo n and studies its basic properties, but the work adds little beyond routine extensions.

read the letter

Dear colleague, the main takeaway is that the authors introduce tilde-varphi(n) as the count of m from 1 to n that are coprime to n but not prime, then examine some of its properties. This is essentially phi(n) with the primes removed from the set of units. The definition itself is clean and uses only standard predicates, so the function is well-formed from the start. They appear to work out its values or behavior in simple cases such as prime powers or small n, which is the sort of explicit calculation that can be checked directly. That part is done competently enough on its own terms. The paper stays within the usual arithmetic-function toolkit and does not claim deep theorems or applications, which keeps the claims modest. The soft spot is that the modification is too direct to generate much new insight. Once you subtract the primes that happen to be coprime to n, most further properties follow immediately from known facts about phi and the primes dividing n; nothing in the abstract suggests a non-obvious identity or connection that would justify the effort. The central statement that the function possesses various properties is true once the function is defined and inspected, but that does not make the inspection itself significant. Because the contribution stays at the level of a straightforward restriction, the paper is mainly of interest to readers who like cataloguing small variants of classical functions. It is unlikely to affect how anyone thinks about units modulo n or to be cited outside a narrow circle of similar work. I would not bring it to a reading group or cite it myself. The thinking is clear and the definitions are honest, yet the results do not rise to the level that would merit referee time.

Referee Report

0 major / 3 minor

Summary. The manuscript defines the arithmetic function tilde-varphi(n) as the number of integers m with 1 ≤ m ≤ n such that gcd(m,n)=1 and m is not prime. It then examines various properties of this function, including explicit formulas, behavior under multiplication, and explicit values or tables for small n.

Significance. The function is obtained from Euler's totient function φ(n) by subtracting the primes p ≤ n that are coprime to n. Any properties derived from standard facts about φ(n) and the prime-counting function are therefore routine. The work supplies no new closed-form expressions, applications, or connections to deeper results in analytic number theory, so its significance remains modest even if all stated identities hold.

minor comments (3)

The abstract and introduction should explicitly relate tilde-varphi(n) to φ(n) minus the count of primes in the unit group, to clarify that the object is a direct restriction rather than an independent construction.
Any tables of computed values should include the corresponding φ(n) and π(n) entries for direct comparison, and state the range of n examined.
If multiplicative properties are claimed, the manuscript must verify them against the standard criterion for arithmetic functions (i.e., check on prime powers and coprime arguments) rather than merely listing examples.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report and the recommendation of minor revision. The referee observes that the defined function is obtained from standard facts about Euler's totient and prime counting, rendering the derived properties routine and the overall significance modest. We respond to this assessment below.

read point-by-point responses

Referee: The function is obtained from Euler's totient function φ(n) by subtracting the primes p ≤ n that are coprime to n. Any properties derived from standard facts about φ(n) and the prime-counting function are therefore routine. The work supplies no new closed-form expressions, applications, or connections to deeper results in analytic number theory, so its significance remains modest even if all stated identities hold.

Authors: We agree that tilde-varphi(n) equals φ(n) minus the count of primes p ≤ n with gcd(p,n)=1, which is equivalent to π(n) minus the number of distinct prime divisors of n. The manuscript derives explicit formulas expressing tilde-varphi(n) in terms of these quantities, establishes its behavior for composite arguments (including multiplicativity when arguments are coprime), and supplies tables of values. While these identities follow from known properties, the manuscript presents a focused, self-contained examination of the count of non-prime units modulo n that has not appeared previously. We make no claim to new closed forms independent of φ and π, nor to applications in analytic number theory; the contribution is the elementary study of this particular arithmetic function. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces the arithmetic function tilde-varphi(n) by direct definition as the count of integers m from 1 to n that are coprime to n but not prime. It then examines standard properties of this function. No derivation chain, equations, fitted parameters, or self-citations are present that reduce any claimed result to the inputs by construction. The work is self-contained as a routine extension of Euler's totient function using elementary predicates, with all properties following from established number-theoretic facts without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies only a definition; no free parameters, axioms, or invented entities are stated or detectable.

pith-pipeline@v0.9.0 · 5565 in / 972 out tokens · 15557 ms · 2026-05-25T01:08:00.149626+00:00 · methodology

Number of non-primes in the set of units modulo n

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)