Assuming that $\zeta(s)$ has no nontrivial zero with $\mathrm{Re}\, s>7/8$, as asserted in a recent unrefereed preprint of OpenAI, we obtain $\Lambda \le 13963/100000 = 0.13963$ for the de Bruijn-Newman constant. A published Polymath boundary computation supplies a finite-region strip of half-width $1/5$. We derive an approximation on $3 \le \mathrm{Im}\, z \le 6$ from its general Riemann-Siegel coefficient bounds. At height $61/20$, a fixed cutoff permits two differentiations, and comparison with $\zeta$ bounds both the zero-density kernel and its vertical derivative. A new conjugate-pair inequality and a twenty-four-step strip comparison give the stated bound. The finite scalar and polynomial certificates are included in the paper; earlier comparisons giving $0.17973$, $0.17172$, and $0.16408$ are retained to show the source of the gain. The argument uses neither Gordon's analytic inputs nor Gomila's new computations. The seven-eighths hypothesis remains an assumption, and the published numerical boundary input is retained.
Let $\lambda_1(\Omega)$ and $T(\Omega)$ denote the first Dirichlet eigenvalue and torsional rigidity of a bounded convex domain $\Omega\subset\mathbb{R}^2$, and let $M=\max_\Omega u$, where $u$ is the torsion function. We prove the sharp inequalities $\pi^2/24<\lambda_1(\Omega)T(\Omega)/|\Omega|<\pi^2/12$. This proves...
Let $E(X)=\#\{(a,b)\in\Z^2:a^2+b^2\le X\}-\pi X$. We prove that $E(X)=O_\varepsilon(X^{209/670+\varepsilon})$ for every $\varepsilon>0$, uniformly for real $X\ge4$. This breaks the $5/16$ threshold suggested by Huxley as a possible limit of the Bombieri--Iwaniec method at the time. The proof introduces two refinements:...
Xi Cen, Xi-Tao Gao, Cheng Zhang et al.· 0 citations
We introduce two independent numerator exponents into the Zeilberger--Zudilin integral and specialize them to \[ A_1=A_2=\frac{1857}{2785}. \] The resulting integer linear forms in $1$ and $\pi$ prove \[ \mu(\pi)<7.101862832357. \] This lowers the Zeilberger--Zudilin upper bound $7.103205334137\ldots$ by more than $0.0...
Suppose a set of integers $A\subseteq\{1,\dots,N\}$ has no solutions to $a-a'=n\lfloor\sqrt[3]2n\rfloor,$ for distinct $a,a'\in A,$ and $n\in \mathbb{N}.$ We show that $|A|\ll N^{1-c}$ for some absolute constant $c>0.$ To do this, we prove quantitative bounds on the van der Corput property for certain sets of bracket q...
Let $\Omega=2\int_0^1(1-t^4)^{-1/2}dt$. We prove that, for every real $\nu>2$, there is $Q\geq2$ such that $q^{-\nu}<|\Omega-p/q|$ for every integer $p$ and every integer $q\geq Q$. Consequently $\Omega$ is irrational and its irrationality exponent is two. The proof combines elliptic separated-weight interpolation, ari...
Let $E_2$ be the extraspecial group of order $3^5$ and exponent three. We prove that $d(E_2\times C_3^r)=2r+10$ for $0\le r\le3$. The upper bounds follow from signed zero-block identities and two finite statements in the four-dimensional symplectic space over $\mathbb F_3$. The first supplies edge weights for all compl...
André Volkmann· 0 citations
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