bibkey: szabo2024primeproducts authors: Barnabás Szabó year: 2024 title: On the existence of products of primes in arithmetic progressions doi: 10.1112/blms.12990 url: https://arxiv.org/pdf/2208.05762v1 claim: Proposition 6 bounds a triangular logarithmically weighted prime-character sum for bounded-order nonprincipal characters; the actual CA application uses all odd exponent layers and retains ramified zeros. strata_touched: [] license: citation-only triage: anchor

A signed prime sum that preserves the actual odd layers

The article appeared in Bulletin of the London Mathematical Society 56 (2024), 1227–1243, DOI:10.1112/blms.12990. The inspected primary sources are the published PDF, Proposition 6 and its proof, printed pp.1233–1234, and arXiv:2208.05762v1, dated 11 August 2022, where Proposition 6 and its proof are on printed p.7. The result is reused without reproving the analytic estimate. No complete proof audit, effective error modulus, Lean verification or strongest-available-result claim is made.

For fixed $\alpha >0$, put $f_{\alpha }(t)=(\alpha -t{)}_{+}$ for $t\ge 0$. Proposition 6 gives, as $q\to \infty$, for nonprincipal Dirichlet characters $\chi \mathrm{mod}q$ of bounded order,

$$\mathrm{Re}\sum _p\frac{\chi (p)\log p}{p}{f}_{\alpha }\left(\frac{\log p}{\log q}\right)\le \left(\frac{\alpha }{8}+o(1)\right)\log q.\text{\hspace{1em}(S1)}$$

The support of the weight is $p\le q^{\alpha }$. The modulus need not be prime, cube-free or bounded in its cubic part; the paper’s separate prime-product theorems have their own conditions. Quadratic characters have the required bounded order two. Primes dividing $q$ have value zero. No GRH or deletion of an exceptional character is imposed in (S1).

The source applies Heath-Brown’s Lemma 5.2 with $s=1$; bounded character order permits $\varphi =1/4$. The Laplace transform satisfies $\mathrm{Re}F(z)\ge 0$ for $\mathrm{Re}z\ge 0$, so zero terms, including a possible exceptional real zero, have the required sign and can be discarded. Higher prime powers cost $O_{\alpha }(1)$. These are the source’s proof ingredients, not a new spectral correspondence between finite FIB frequencies and zeta zeros.

At each fixed $\alpha$ and fixed order bound two, the error in (S1) is uniform over the varying quadratic characters and moduli. The character-independent threshold is explicit in the restatement of the referenced lemma in Xylouris, arXiv:0906.2749v1, Lemma 2.2, printed p.15: its $q_0(f,\epsilon )$ and auxiliary radius do not depend on the character. That restatement and the source’s use of the lemma were inspected; the original 1992 analytic proof was not independently audited.

The actual CA application uses (S1) with a triangular weight transported from the scale $P^{+}(n)$. It retains the same integer’s character, all tied choices and conductor zeros. It does not infer reciprocal-prime mass or a signed Robin margin merely from this signed logarithmic weight.