Real-Variable Proof of the Nyman Criterion

Definitions and mathematical statement

For real $x\geq 2$, let $\pi(x)=\#\{p\leq x:p\text{ is prime}\}$, and let $\operatorname{li}(x)=\operatorname{PV}\int_0^x du/\log u$ be the logarithmic integral. For $u\in{\mathbb{R}}$, write $\{u\}=u-\lfloor u\rfloor$. For $t>0$ and real $\alpha>1$, define \[ \chi(t)=\begin{cases}1,&t>1,\\0,&0<t\leq 1,\end{cases} \qquad e_\alpha(t)=\{t/\alpha\}. \] The first assertion in the supplied statement is \[ \tag{A} \forall\varepsilon>0\ \exists C_\varepsilon>0\ \exists X_\varepsilon\geq2\ \forall x\in{\mathbb{R}},\ x\geq X_\varepsilon: \quad |\pi(x)-\operatorname{li}(x)|\leq C_\varepsilon x^{1/2+\varepsilon}. \] The source writes the second assertion using a sum $\sum_\alpha c_\alpha e_\alpha$, without explicitly specifying its support or coefficient field. Under the finite-linear-combination interpretation, its precise formulation, for a specified field $K\in\{{\mathbb{R}},{\mathbb{C}}\}$, is \[ \tag{B$_K$} \begin{gathered} \forall\delta>0\ \exists n\in{\mathbb{N}},\ n\geq1,\ \exists\alpha_1,\ldots,\alpha_n\in(1,\infty)\ \exists c_1,\ldots,c_n\in K:\\ \int_0^\infty\left|\chi(t)-\sum_{j=1}^n c_j\{t/\alpha_j\}\right|^2\frac{dt}{t^2}<\delta. \end{gathered} \] The parameters and coefficients may depend on $\delta$. The integral is with respect to the weighted Lebesgue measure $t^{-2}\,dt$. No additional coefficient constraint is stated in the supplied record.

The requested task is to give a real-variable proof of the equivalence of (A) and the intended second assertion. With the explicitly qualified finite-sum interpretation above, this is the equivalence of (A) and $\mathrm{(B}_K\mathrm{)}$, after the coefficient field has been specified. The finite-support interpretation and coefficient field require confirmation from the cited report; an infinite-sum interpretation would additionally require a convergence convention. The supplied record also does not define which methods qualify as a real-variable proof. Its original wording refers to assertions (2) and (3) of the report; the expanded assertions above come from the supplied clean statement, without independent verification of that correspondence. The target is a proof with the requested methodological restriction, rather than a request to establish (A) unconditionally.

Short English statement

Find a real-variable proof that the stated prime-counting error estimate is equivalent to arbitrarily accurate approximation of $\chi$ by combinations of the fractional-part functions $e_\alpha$ in $L^2((0,\infty),t^{-2}\,dt)$. The displayed finite-sum formulation is explicitly conditional on that interpretation of the source. The coefficient field and the permitted proof methods require clarification. The supplied open label concerns the requested proof method.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002495$, title Real-Variable Proof of the Nyman Criterion, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives the proposed year as 2014 and identifies no individual proposer.
• Dirichlet Series and Function Theory in Polydiscs (2014), Oberwolfach Reports, pp. 335--393, DOI 10.4171/owr/2014/06. Supplied source report and literature-source link: https://doi.org/10.4171/owr/2014/06.
• Supplied literature-source label: "Baez-Duarte real-variable reformulation," https://doi.org/10.48550/arXiv.math/0011254. No further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the methodological question open, reports that existing proofs use analytic or arithmetic machinery, and states that no such real-variable proof was found. This is supplied dated curation, not an independently verified literature assessment. The background expressly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and states that it was checked against the official Oberwolfach report. This is a supplied verification claim, not a source check performed for this draft.