CFKRS Formula for Shifted Zeta Function Moments

Definitions and mathematical statement

The supplied record attributes to the CFKRS conjecture a power-saving asymptotic for smoothed products of shifted Riemann zeta-functions. Here $\zeta(s)=\sum_{n\geq1}n^{-s}$ for $\Re s>1$, with its meromorphic continuation elsewhere.

The record does not define $\tau_C$. A conventional interpretation, stated here only to explain the notation, is that for a finite multiset $C=\{c_1,\ldots,c_r\}$ of complex shifts, \[ \tau_C(n)=\sum_{n_1\cdots n_r=n}\prod_{j=1}^{r}n_j^{-c_j}, \qquad \prod_{c\in C}\zeta(s+c)=\sum_{n\geq1}\frac{\tau_C(n)}{n^s} \] in a half-plane of absolute convergence. For the empty multiset, the corresponding coefficients are $\tau_{\varnothing}(1)=1$ and $\tau_{\varnothing}(n)=0$ for $n>1$. Under this convention, the supplied Dirichlet-series factor is \[ B(C,D)=\sum_{n\geq1}\frac{\tau_C(n)\tau_D(n)}{n}, \] where this series converges absolutely. Its meaning outside that region requires a continuation or regularization prescription that is absent from the record.

The supplied conjectural assertion has only the schematic form \[ \mathcal M(T) = \sum_{\substack{U\subseteq A,\ V\subseteq B\\ |U|=|V|}} \mathcal S_{U,V}(T) +O\!\left(T^{1-\delta}\right), \qquad T\to\infty, \] with a positive power-saving exponent $\delta$. In this display, $A,B$ are placeholders for the unspecified collections of small complex shifts, $\mathcal M$ denotes the unspecified smoothed zeta-product average, and $\mathcal S_{U,V}$ denotes the unspecified term associated with an equal-cardinality swap. These symbols expose the structure described by the record; they do not supply the missing formula. The smoothing, averaging domain, zeta arguments, admissible shifts, swap operation, main-term factors, and dependence and uniformity of the error are not specified. Consequently, the supplied evidence does not determine a fully quantified mathematical conjecture.

There is also an unresolved target discrepancy: the statement concerns shifted zeta-function products, whereas the supplied literature assessment concerns products of quadratic $L$-functions. These formulations cannot be identified from the record alone.

Short English statement

The record describes a conjectural formula for smoothed shifted zeta moments: add all equal-cardinality swap contributions and obtain an error of size $O(T^{1-\delta})$ for some positive $\delta$. Essential definitions and quantifiers are missing, and the accompanying literature assessment discusses quadratic $L$-functions instead. A precise complete formulation requires source review.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30004288$, source problem code catalogue entry, titled "CFKRS Formula for Shifted Zeta Function Moments." No proposer is identified. The recorded proposed year is 2019.
• Supplied report citation: Analytic Number Theory (2020), Oberwolfach Reports, pp. 3141--3205, DOI 10.4171/owr/2019/50. https://doi.org/10.4171/owr/2019/50. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No precise location for this conjecture within the report is supplied.
• Supplied literature source: Conrey--Rodgers, "quadratic twists of long Dirichlet polynomials." https://doi.org/10.1007/s00209-025-03848-x.
• Supplied literature source: Conrey--Rodgers, "preprint and stated range." https://arxiv.org/abs/2208.01783.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2019/50.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. It labels the problem open and states that the shifted product-of-quadratic-$L$-functions recipe remains conjectural in full generality, with available proofs covering only restricted ranges or conditional special cases. Its applicability to the zeta-moment statement is unresolved. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, but its verification note limits the changes to formula delimiters, notation, grammar, and typography while preserving mathematical content and scope. This label does not establish mathematical correctness or completeness.