bibkey: revesz1988oscillation
authors: Szilárd Gy. Révész
year: 1988
title: Effective oscillation theorems for a general class of real-valued remainder terms
doi: 10.4064/aa-49-5-481-505
url: https://doi.org/10.4064/aa-49-5-481-505
claim: The proof following Lemma 6 supplies a discrete real-power-sum fact, and Lemma 4 records the classical low-ordinate zero boundary; these facts are used in a signed Gaussian application to the complete Mertens primitive, without treating the paper's unsigned remainder theorem as a signed primitive theorem.
strata_touched: []
license: citation-only
triage: anchor

A finite power-sum selection and its signed-primitive application

The primary is Acta Arithmetica 49(5), 481–505. The publisher’s open download resolves to the ICM scan, 883,715 bytes, 13 PDF pages, SHA-256 01c8e8a41018223d30c841f48282755502ca48387f4cfbf2eb852990704d420c. Printed pp.481–491 were visually inspected; the remaining proof was not independently audited or Lean verified.

The proof following Lemma 6, printed p.488, uses the standard discrete power-sum fact for roots of a real polynomial: one of the first $d+1$ power sums of a degree-$d$ real polynomial is nonnegative. Applying that fact to any $n-1$ complex numbers together with their conjugates, and adjoining a designated number $1$, gives the loose integer-index consequence

$$z_1=1⟹\underset{1\le \nu \le 2n+1}{\max }\Re \sum _{j=1}^n{z}_j^{\nu }\ge 1.$$

The $n=1$ case is immediate. This is a consumed application of the existing Cassel step, not a new power-sum theorem. No modulus restriction is imposed on the other $z_j$. Multiplicities remain in the sum. Lemma 4, printed p.486, also records the first ordinates of the classical Riemann zeros; the only consequence used is that all nontrivial zeros have ordinate of absolute value greater than $10$.

The complete positive-primitive localization, §479 uses this finite selection in the actual integrated zero spectrum and pays the nonoscillatory polynomial-Gaussian envelope and every omitted tail. Its explicit positive-primitive and literal-lcm conclusions are repository paper applications, not conclusions stated in Révész’s article. The source’s Theorem 1 is an absolute-value remainder estimate in its stated logarithmic-derivative class; it is not substituted for that signed application. Neither the source facts nor this application construct an actual off-critical zero or prove RH.

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• DOI: https://doi.org/10.4064/aa-49-5-481-505