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Source. Display (9) and the identity before it, p. 3, in §2 "Prime-power residue families" of A Two-Copy Proof of Erdős Problem 126 (2026), a three-page preliminary exposition with no printed author, posted at https://www.erdosproblems.com/static/126-proof.pdf; the edition read is identified on the source card. The step is unlabelled in the print; this page names it.

Statement

For finitely many positive real numbers $x_i$, the kernel $L(i,j)=\log(x_i+x_j)$ is conditionally negative semidefinite:

$$\sum_{i,j}z_iz_jL(i,j)\leq0 \quad\text{whenever}\quad \sum_i z_i=0.$$

No sign is asserted for vectors whose coordinates do not sum to zero.

Read depth. Claims checked: the identity and display (9) were read on p. 3, and the convergence remark below was added here. Nothing here is independently reviewed.

Proof sketch

P. 3. With $\rho_i=(x_i-1)/(x_i+1)$, which lies in $(-1,1)$,

$$\log(x_i+x_j)=\log(x_i+1)+\log(x_j+1)-\log2+\log(1-\rho_i\rho_j).$$

For a zero-sum $z$ the first three terms drop out of the quadratic form, and expanding $\log(1-t)=-\sum_{m\geq1}t^m/m$ writes the remainder as $-\sum_{m\geq1}\frac1m\bigl(\sum_iz_i\rho_i^m\bigr)^2\leq0$, which is (9). The print does not justify the rearrangement; it is valid because the $m$-th term is at most $(\sum_i|z_i|)^2q^{2m}/m$ in absolute value, with $q=\max_i|\rho_i|<1$.

Dependencies

The power series of $\log(1-t)$ for $|t|<1$.

Used by. [[arithmetic_functions/adamczewski_2026_erdos126/main_theorem|The main theorem]], with $x_i=a_i$.

Bears on

• Problem 126: the lemma supplies the conditional negativity that the main theorem needs to apply Proposition 1; it bears on the problem only through that theorem.