Mean value theorems for a class of density-like arithmetic functions

Lucas Reis

Mean value theorems for a class of density-like arithmetic functions

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves a mean value theorem for divisor-product arithmetic functions, and uses it to show that the densities of primitive and normal elements in finite field extensions have positive average values.

desk verdict Genuinely new mean value theorem for divisor-product functions, with the normal-element density result as a clean payoff; minor omissions, no load-bearing flaws. read the letter →

arxiv 1908.01198 v2 pith:A7WI5S7T submitted 2019-08-03 math.NT

Lucas Reis This is my paper · ORCID

classification math.NT MSC 11H6011N3711T30

keywords meanvaluetheoremarithmeticfunctionsnormalelementsprimitivefinitefieldsdensity-likedivisor-productEulertotientconvolution

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a mean value theorem for arithmetic functions of the form $f(n)=\prod_{d|n}g(d)$, where $g$ takes values in $(0,1]$ and only needs the series $\sum_{d\ge1}(1-g(d))/d$ to converge. It shows that such an $f$ always has a long-run average, or mean value, computed as the limit of a sequence built from Euler totient weights, and that the mean value is positive whenever $g$ stays bounded away from zero. This matters because the densities of primitive and normal elements in the finite-field extension $\mathbb{F}_{q^n}$ both have exactly this divisor-product shape. Consequently, both densities have positive average values, the normal-element average tends to $1$ as $q$ grows, and the primitive-element average has $ liminf$ equal to $0$ across $q$.

What carries the argument

The engine is the notion of an $N$-density-like function: $g$ takes values in $(0,1]$ and equals $1$ whenever $\gcd(n,N)>1$, so only divisor arguments built from primes outside $N$ are allowed to lower the product. The averaging sequence $A_t=\frac1{L_t}\sum_{r|L_t}f(r)\varphi(L_t/r)$ is then shown to be nonincreasing in $t$, and any discrepancy between the partial average of $f(n)$ and $A_t$ is bounded by the tail of $\sum(1-g(d))/d$. For normal elements, the additional machinery is the polynomial Euler totient: the count of normal elements in $\mathbb{F}_{q^n}$ equals $\Phi_q(x^n-1)$, and cyclotomic factorization turns this count into the divisor product $\prod_{d|n}(1-q^{-e_q(d)})^{\varphi(d)/e_q(d)}$, which is exactly the form the main theorem needs.

What would settle it

Fix a small prime power $q$ and compute the partial averages $x^{-1}\sum_{n\le x}\mu_q(n)$ alongside the sequence $A_t$ from Theorem 4.2; if the two limits differ, or if for some $n$ the number of normal elements of $\mathbb{F}_{q^n}$ is not $\Phi_q(x^n-1)$, the claimed positive mean value fails.

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Extended reading notes

Core claim

The central discovery is that the average behavior of a divisor-product function $f(n)=\prod_{d|n}g(d)$ is controlled by the convergence of $\sum(1-g(d))/d$ and by the values $g$ takes away from a fixed set of primes. Under those conditions the partial averages $(1/x)\sum_{n\le x}f(n)$ converge to a limit $A_f$, and $A_f$ can be computed through the totient-weighted sequence $A_t=\frac1{L_t}\sum_{r|L_t}f(r)\varphi(L_t/r)$ for any chain $L_1|L_2|\cdots$ of integers that eventually contains every integer relatively prime to $N$. If, in addition, $g(d)>c>0$ for all $d$, then $A_f>0$ and in fact $A_f\ge\prod_{d\ge1}g(d)^{1/d}$. Applied to finite fields, the theorem shows that the average density of primitive elements and the average density of normal elements over extensions $\mathbb{F}_{q^n}$ are both positive, with the normal average satisfying $1-\frac1q-\frac1{\sqrt q}<\mu_q\le1-\frac1q$ for $q\ge4$.

Load-bearing premise

The application to normal elements rests on the cited identity that the number of normal elements in $\mathbb{F}_{q^n}$ equals $\Phi_q(x^n-1)$; if that identity failed, $\mu_q(n)$ would not factor as a divisor product and the main theorem would not apply.

Editorial extensions

If this is right

Any divisor-product $f$ built from a $(0,1]$-valued $g$ with convergent $\sum_{d\ge1}(1-g(d))/d$ has a finite mean value, so questions about such functions reduce to checking one series.
If $g(d)>c>0$ uniformly, the mean value is positive, with the explicit lower bound $A_f\ge\exp(\sum_{d\ge1}\log g(d)/d)$.
All moments of $f$ have mean values, so $f$ has a finite variance whenever the second-moment limit exists.
The normal-element density has mean $\mu_q$ with $1-\frac1q-\frac1{\sqrt q}<\mu_q\le1-\frac1q$ for $q\ge4$, hence $\mu_q\to1$ as $q\to\infty$.
For $q\ge4$ and any $T>0$, all but at most $x/(1+T\sqrt q)$ positive integers $n\le x$ satisfy $\mu_q(n)\ge1-\frac1q-\frac1{\sqrt q}-T$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because $N=1$ is allowed, the theorem is effectively a statement about every $(0,1]$-valued $g$ whose series $\sum(1-g(d))/d$ converges; the $N$-density-like condition is a bookkeeping device that lets the proof focus on finitely many primes.
The proof's monotonicity of $A_t$ suggests that if the tail of $\sum(1-g(d))/d$ has a known decay rate, the $o(1)$ error in the mean-value statement can be made quantitative, giving explicit rates of convergence rather than just existence.
The same divisor-product mechanism would plausibly apply to other finite-field statistics with totient-analogue factorizations, such as elements with prescribed multiplicative order or prescribed trace, producing positive average densities for those families.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The real news here is Theorem 3.2, a general mean value theorem for f(n)=∏_{d|n} g(d) where g: N→(0,1] is “N-density-like” and Σ(1-g(d))/d converges. That statement is not in the cited literature, and the proof is self-contained: divisor averaging along a chain L_t, monotonicity of A_t, and a tail-sum argument for the S_2 term. I checked the monotonicity step in Lemma 3.6 and the S_2 bound in Proposition 3.5; both work. The AM-GM argument for Af ≥ exp(A*_f) is also sound. This is a solid, useful addition, not a rehash of Wintner–Wirsing.

The applications are where the paper earns its keep. The primitive-element density mean value was already known from Shparlinski, but the proof here is different and the lim inf result is new. The normal-element result is the highlight: μ_q(n), the density of normal elements in F_{q^n}, has a positive mean value, with explicit bounds 1 − 1/q − 1/√q < μ_q ≤ 1 − 1/q for q ≥ 4. The variance estimate in Corollary 4.9 is a nice extra. The reduction of normal-element counting to the classical identity Φ_q(x^n−1) is standard and correctly applied.

Soft spots are minor. Lemma 2.1 is stated without proof, but it is a routine inequality. In Theorem 4.8, the check for q^j ≤ 10^4 is left as a direct computation; it is finite and not load-bearing. Lemma 4.7's estimate 1−G_q(d) ≤ log q / log(d+1) is slightly compressed—it uses e_q(d) ≥ log_q(d+1), which follows from q^{e_q(d)} ≡ 1 mod d—but it checks out. The paper's citation pattern is clean; no fitted parameters, no circularity. The main theorem's value extends beyond finite fields, so I'd expect it to be cited.

This paper deserves a serious referee. It is coherent, the proofs are detailed, and the main results are new. I would accept it after minor revision, mostly for the paper to spell out the omitted finite check and the Lemma 4.7 inequality. Bring it to reading group; it is a good example of a simple averaging argument with clean applications.

Referee Report

0 major / 6 minor

Summary. The paper studies arithmetic functions of the form f(n)=∏_{d|n} g(d), where g takes values in (0,1] and is N-density-like, and proves a mean value theorem under the condition that Σ (1-g(d))/d converges. Theorem 3.2 shows that f has a mean value A_f, computed as the limit of A_t=(f*φ)(L_t)/L_t for any admissible ladder L_t; if g is bounded below by a positive constant, A_f>0 and the logarithmic mean value is also identified. The proof proceeds by an approximating-average step (Proposition 3.5) and a monotonicity step (Lemma 3.6), with the series condition controlling the error. The finite-field applications show that the primitive-element density ρ_q(n)=φ(q^n-1)/q^n and the normal-element density μ_q(n)=Φ_q(x^n-1)/q^n have positive mean values ρ_q and μ_q; Theorem 4.8 gives 1-1/q-1/√q < μ_q ≤ 1-1/q for q≥4, and Corollaries 4.9 and 4.10 give variance and distributional consequences. The paper also proves liminf_{q→∞} ρ_q=0 and lim_{q→∞} μ_q=1.

Significance. If the main theorem holds, this is a genuinely useful addition to the mean-value toolbox: it applies to non-multiplicative functions, requires only a natural one-variable series condition, and yields a computable limiting value together with positivity. The finite-field applications are well chosen, and the normal-element quantitative bounds appear to be new and of independent interest. The proofs are detailed and checkable; the monotonicity argument for A_t and the error control by the convergent series are correct. The paper relies on two standard external results, the normal-element count of Lidl-Niederreiter (Theorem 4.5) and the Nicolas-Robin divisor bound (Lemma 2.2), and on the main theorem's own hypotheses rather than on its conclusion. There is no circularity and no fitting of constants. The finite check in Theorem 4.8 should be made explicit, but it does not affect the central existence theorems.

minor comments (6)

[§2.1, Lemma 2.1] The lemma is stated without proof but is used repeatedly in the main estimates, including Eq. (3.1) and Lemma 3.7; since the proof is a short Bernoulli/Bonferroni-style induction, I recommend adding it or an explicit citation.
[§4.2.1, Lemma 4.7] The statement reads 'Let G_q(d) be as in Eq. (4.1)', but G_q is defined in Eq. (4.3); this cross-reference should be corrected.
[§4.2.1, proof of Theorem 4.2] In the paragraph after Eq. (4.3), the sentence 'in order to prove that ρ_q(n) has positive mean value' should refer to μ_q(n), not ρ_q(n).
[§4.2.2, Theorem 4.8] The finite verification 'By a direct computation, we verify that the same holds in the range q^j≤10^4' is not shown in the paper; because this check supports the quantitative lower bound for μ_q, please provide the details, a short table, or a small verification script.
[§3, Theorem 3.2, condition (3)] The condition 'h(t)<n for every integer n≥1 not dividing L_t' uses the symbol n both as the dummy integer and as the argument of f(n) in the same theorem; renaming the dummy (for example m) would avoid ambiguity.
[§1, Introduction] There is a typo in the Introduction: 'good estiamtes' should be 'good estimates'.

Circularity Check

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No significant circularity: the mean value theorem is derived from first principles and the finite-field applications use independent, standard external results as ingredients.

full rationale

The paper's central theorem (Theorem 3.2) is proved directly: it constructs the candidate mean value as a limit of the averaged convolution (f * phi)(L_t)/L_t and proves convergence using the monotonicity lemma (Lemma 3.6), the tail estimate from convergence of Sum (1 - g(d))/d, and the density-like conditions on g. The proof is self-contained and does not assume the mean value it derives. The applications are also not circular. For primitive elements, the identity f_q(n) = phi(q^n - 1)/(q^n - 1) = product over d|n of g_q(d) is a reformulation of the definition of multiplicative order, and the convergence of Sum (1 - g_q(d))/d is established via the independent Nicolas-Robin divisor bound. For normal elements, the count of normal elements is quoted from a standard external reference (Lidl-Niederreiter, Theorem 3.73), and the factorization mu_q(n) = product G_q(d) follows from the cyclotomic factorization and the standard Lemma 4.6. This external identity is an ingredient, not a conclusion derived from the paper's own framework; the same density-like theorem is then applied with independent hypotheses. The only omitted check is an explicit finite-range verification in Theorem 4.8, which is not load-bearing for the existence or positivity of the mean values and does not amount to fitting the conclusion into the proof. No fitted parameter is renamed as a prediction, no load-bearing self-citation appears, and no known result is merely renamed. The derivation chain is therefore self-contained in the sense relevant to circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard finite field factorization results and a divisor function bound, all cited from prior literature. No free parameters are fitted and no new entities are introduced.

assumptions (3)

domain assumption The number of normal elements in F_{q^n} over F_q equals Φ_q(x^n-1) (Theorem 4.5, cited to [3], Section 4.2).
This identity is the foundation for the product formula for µ_q(n); without it the application of Theorem 3.2 to normal elements collapses.
domain assumption Cyclotomic polynomial E_d(x) factors into φ(d)/e_q(d) distinct irreducibles of degree e_q(d) over F_q (Lemma 4.6, cited to [3]).
Used to compute the polynomial Euler totient of x^n-1 and obtain the explicit form of G_q(d).
standard math The divisor function bound σ_0(m) < m^{1.1/log log m} for m≥3 (Lemma 2.2, cited to [4]).
Used in Lemma 4.7 to prove convergence of the series for normal elements.

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Cite this review

Pith. "Pith review of Mean value theorems for a class of density-like arithmetic functions." pith.science (2026). https://pith.science/paper/A7WI5S7T

@misc{pith2026190801198,
  author       = {Pith},
  title        = {Pith review of: Mean value theorems for a class of density-like arithmetic functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A7WI5S7T}},
  note         = {Machine review of arXiv:1908.01198}
}

abstract

This paper provides a mean value theorem for arithmetic functions $f$ defined by $$f(n)=\prod_{d|n}g(d),$$ where $g$ is an arithmetic function taking values in $(0, 1]$ and satisfying some generic conditions. As an application of our main result, we prove that the density $\mu_q(n)$ (resp. $\rho_q(n)$) of normal (resp. primitive) elements in the finite field extension $\mathbb{F}_{q^n}$ of $\mathbb{F}_q$ are arithmetic functions of (non zero) mean values.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Diﬃe and M

W. Diﬃe and M. Hellman. New directions in cryptography. IEEE Trans. Information Theory, 22 (1976) 644 –654

work page 1976
[2]

Frandsen, On the density of normal bases in ﬁnite ﬁel ds, Finite Fields Appl

G.S. Frandsen, On the density of normal bases in ﬁnite ﬁel ds, Finite Fields Appl. 6 (2000) 23–28

work page 2000
[3]

Lidl and H

R. Lidl and H. Niederreiter, Introduction to ﬁnite ﬁelds and their applications , (Cam- bridge University Press New York, 1986)

work page 1986
[4]

L.Nicolas and G

J. L.Nicolas and G. Robin, Majorations explicites pour l e nombre de diviseurs de N , Can. Math. Bull. 26 (1983) 485–492

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[5]

Gao, Normal basis over ﬁnite ﬁelds , (PhD thesis, University of Waterloo, 1993)

S. Gao, Normal basis over ﬁnite ﬁelds , (PhD thesis, University of Waterloo, 1993)

work page 1993
[6]

Gao and D

S. Gao and D. Panario Density of Normal Elements, Finite Fields Appl. 3 (1997) 141–150

work page 1997
[7]

J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers Illiinois J. Math. 6 (1962) 64–94

work page 1962
[8]

Shparlinski On some arithmetic properties of recurre nt sequences, Math

I. Shparlinski On some arithmetic properties of recurre nt sequences, Math. Notes (in Russian) 47 (1990) 612-617

work page 1990

Show all 10 references

[9]

Wintner, The theory of measure in arithmetical semigroups (Baltimore, 1944)

A. Wintner, The theory of measure in arithmetical semigroups (Baltimore, 1944)

1944
[10]

Wirsing, Das asymptotische Verhalten von Summen ¨ ub er multiplikative Funktio- nen II, Acta Math

E. Wirsing, Das asymptotische Verhalten von Summen ¨ ub er multiplikative Funktio- nen II, Acta Math. Acad. Sci. Hung 18 (1967), 411–467

1967

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Diﬃe and M

W. Diﬃe and M. Hellman. New directions in cryptography. IEEE Trans. Information Theory, 22 (1976) 644 –654

work page 1976

[2] [2]

Frandsen, On the density of normal bases in ﬁnite ﬁel ds, Finite Fields Appl

G.S. Frandsen, On the density of normal bases in ﬁnite ﬁel ds, Finite Fields Appl. 6 (2000) 23–28

work page 2000

[3] [3]

Lidl and H

R. Lidl and H. Niederreiter, Introduction to ﬁnite ﬁelds and their applications , (Cam- bridge University Press New York, 1986)

work page 1986

[4] [4]

L.Nicolas and G

J. L.Nicolas and G. Robin, Majorations explicites pour l e nombre de diviseurs de N , Can. Math. Bull. 26 (1983) 485–492

work page 1983

[5] [5]

Gao, Normal basis over ﬁnite ﬁelds , (PhD thesis, University of Waterloo, 1993)

S. Gao, Normal basis over ﬁnite ﬁelds , (PhD thesis, University of Waterloo, 1993)

work page 1993

[6] [6]

Gao and D

S. Gao and D. Panario Density of Normal Elements, Finite Fields Appl. 3 (1997) 141–150

work page 1997

[7] [7]

J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers Illiinois J. Math. 6 (1962) 64–94

work page 1962

[8] [8]

Shparlinski On some arithmetic properties of recurre nt sequences, Math

I. Shparlinski On some arithmetic properties of recurre nt sequences, Math. Notes (in Russian) 47 (1990) 612-617

work page 1990