Turán-Type Reformulations for Shifted Primes

Definitions and mathematical statement

The source asks for a shifted-prime analogue of "Turán's reformulation" of an additive-function monotonicity problem. It does not state the reformulation, the original theorem, the shift, or the exact logarithm characterization mentioned in the literature assessment. Those missing statements cannot be replaced by a guessed classical theorem.

In the standard real-valued setting, an additive arithmetic function is $f:{\mathbb{N}}\to{\mathbb{R}}$ with \[ f(mn)=f(m)+f(n)\qquad\text{when }\gcd(m,n)=1. \] This is distinct from complete additivity without the coprimality restriction. For a fixed integer shift $a$, define \[ \mathcal S_a=\{p+a\in{\mathbb{N}}:p\text{ prime}\}. \] A usual literal monotonicity condition on these values would be $f(p+a)\leq f(q+a)$ for eligible primes $p<q$. The source does not choose that condition over eventual monotonicity or an asymptotic variant, nor specify whether one or several shifts are involved. The displayed definitions identify possible ambient data rather than asserting the missing theorem.

The request is to find an analogue of the particular Turán reformulation for the intended shifted-prime logarithm-characterization problem. To make that into a proposition one must supply the exact original reformulation, its quantified additive-function class, the shifted-prime hypothesis, and the conclusion's domain and normalization. The record supplies none of those formulas, so no new equivalent criterion or logarithmic identity is invented here.

The supplied literature triage dated 21 August 2026 says the report gives no solution and treats the general reformulation as a research question. It cites a Hildebrand shifted-prime paper but does not reproduce its theorem or identify which Turán statement should be transferred. The broad request is preserved with this substantive formulation gap.

Short English statement

The source seeks a shifted-prime analogue of a particular Turán reformulation but omits the reference theorem and exact hypotheses. Additivity and a possible shifted-prime domain are defined, while the shift, monotonicity convention, and logarithm conclusion remain explicit missing data.

Sources

• Local UnsolvedMath registry, permanent ID $30000134$, exact source code catalogue entry, Oberwolfach Reports Open Problems. Statement status corrected_verified; verification only removes a label and normalizes name typography.
• "Theory of the Riemann Zeta and Allied Functions (2005)," Oberwolfach Reports, pp. 2419--2490, DOI 10.4171/owr/2004/46: https://doi.org/10.4171/owr/2004/46. Proposed year and DOI refer to 2004, while the citation label says 2005.
• Supplied "Hildebrand shifted-prime paper": https://doi.org/10.1112/plms/s3-59.2.209. The other supplied source is the original report; no full statement or separate citation of Turán's reformulation is provided.
• Literature checked 21 August 2026; supplied status open. Registry created 25 August 2026 and updated 16 September 2026. This dated curation triage requires independent verification before citation.
• Registry curation attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Underlying report and literature retain their supplied provenance; this statement-only draft requires human review.