Bretèche–Tenenbaum: Mean values of arithmetic functions and application to sums of powers

../

theorem_1_1: De la Bretèche and Tenenbaum's theorem that, for t at least 2, positive coefficients and non-decreasing exponents with first exponent 2, second exponent 3 or 4, reciprocals of the other exponents summing to one half and conditions (i) to (iii), the integers up to x represented by the sum of powers number at least a constant times x/(log log x)^(5/2).

theorem_3_1: De la Bretèche and Tenenbaum's t-variable counterpart of Henriot's bound: for a function F of the class M_k(A, B, ε) and primitive polynomials Q_j, the sum of F(|Q_1(n)|, …, |Q_k(n)|) over a box of sides y_j is at most a constant times the box volume, a local density sum E_R and a sieve product over primes between g and x.

---

The copy read for this card is arXiv:2403.19320v6 (4 August 2025), 10 pages (the Math. Proc. Camb. Phil. Soc. version was not read). The arXiv abstract page (https://arxiv.org/abs/2403.19320v6) names arXiv's non-exclusive distribution license, and the preprint, stamped "arXiv:2403.19320v6 [math.NT] 4 Aug 2025", prints no notice, every other right reserved.

Régis de la Bretèche, Gérald Tenenbaum, "Mean values of arithmetic functions and application to sums of powers," Math. Proc. Camb. Phil. Soc. 180 (2026), no. 1, 1-13. DOI 10.1017/S0305004125101382.

Overview

The paper studies mean values of arithmetic functions evaluated at polynomial arguments and applies them to integers represented by sums of powers. Theorem 3.1 (p. 4; (3.1)–(3.4)) bounds a multidimensional box sum of a nonnegative function $F(|Q_1(\boldsymbol n)|,\ldots,|Q_k(\boldsymbol n)|)$ satisfying the coprime growth condition (2.1) (p. 3). Its bound combines a product of local sieve factors with $E_{\boldsymbol R}(\mathfrak s\boldsymbol x)$, the sum (3.4) of $\widehat F$ weighted by the local densities built from (2.7) (p. 3). The coefficient and box size restrictions are part of the theorem. The proof uses polynomial congruence bounds (Lemmas 4.1–4.2, pp. 4–5) and a combinatorial sieve estimate (Lemma 4.3, pp. 5–6).

For $r(n)$, the number of representations $n=\sum_{j=0}^t c_jm_j^{\ell_j}$, Theorem 1.1 (p. 2; (1.3)) proves $\#\{n\leq x:r(n)>0\}\gg x/\mathfrak S(x)\gg x/(\log_2 x)^{5/2}$ for $t\geq2$, coefficients $c_j\geq1$ and the specified non-decreasing exponent tuples: $\ell_0=2$, $\ell_1\in\{3,4\}$, $\sum_{j=1}^t1/\ell_j=1/2$, and conditions (i)–(iii). Examples include $(2,3,6)$ and $(2,4,4)$. The proof uses Cauchy–Schwarz (1.5) (p. 2), bounds the unequal first-coordinate contribution through Theorem 3.1 and divisor concentration (Proposition 5.1, pp. 6–8), and handles equal first coordinates by induction using Proposition 5.2 (p. 6; §5.4, p. 9). The bound for three equal terminal powers with exponent at least 26 invokes Salberger’s published results; the suggested threshold 16 rests on private communication (p. 2), not the stated theorem. The upper bound for $\mathfrak S(x)$ in (1.1) (p. 1) is cited background.

Bears on. Problem 301: the paper does not mention unit fractions or the problem. Problem 301 asks for the largest subset of $\{1,\ldots,N\}$ with no identity $1/a=\sum_i1/b_i$ among distinct members; the paper's results bound mean values of arithmetic functions at polynomial arguments and count integers that are sums of powers, and give no bound on that quantity.

Results.

• Theorem 1.1 (p. 2): $V_0(x;\boldsymbol c,\boldsymbol\ell)\gg x/\mathfrak S(x)\gg x/(\log_2x)^{5/2}$ for $t\geq2$, $\ell_0=2$, $\ell_1\in\{3,4\}$, $\sum_{j=1}^t1/\ell_j=\tfrac12$ and conditions (i) to (iii).
• Theorem 3.1 (p. 4): the upper bound (3.3) for sums of $F(|Q_1(\boldsymbol n)|,\ldots,|Q_k(\boldsymbol n)|)$ over boxes in $t$ variables.

Read status: claims checked for Theorems 1.1 and 3.1, the notation of §2, Lemmas 4.1 to 4.3 and Propositions 5.1 and 5.2, read clause by clause on the page images of arXiv:2403.19320v6; the proofs were followed for structure only. Nothing here is independently reviewed.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.