arXiv subjects
Publications and source records attributed to Yajun Zhou.
We prove some conjectures of K. C. Au concerning the descents of cyclotomic levels in certain sums over multiple polylogarithms. Via answers to a question of Au in some particular cases, we also confirm Broadhurst's conjecture on honorary multiple zeta values at even weights.
With generalized central binomial coefficients $ \binom{2x}{x}:=\frac{Γ(2x+1)}{[Γ(x+1)]^2}$ defined through Euler's gamma function, we represent deformed Apéry-like series \[ \mathscr A_{s,n}:=\sum_{k=1}^\infty\left.\!\frac{\partial^n}{\partial x^n}\frac{1}{x^s\binom{2x}{x}}\right|_{x=k} \] by multiple Clausen values (MCVs), which belong to a special class of cyclotomic multiple zeta values (CMZVs) at level $3$. For example, exploiting provable algebraic relations among MCVs, we show that \[\mathscr A_{1,5}=-\frac{9[495L(χ_{-3},6)-30π^{2}L(χ_{-3},4)-2π^{4}L(χ_{-3},2)]}{4}\]and\[\mathscr A_{4,4}=\frac{352ζ_{5,3}}{15}+\frac{752537π^{8}}{10206000},\]where $ L(χ_{-3},s):=\sum_{n=0}^\infty\left[(3n+1)^{-s}-(3n+2)^{-s}\right]$ and $ζ_{5,3}:=\sum_{m>n>0}m^{-5}n^{-3}$.
We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to reduce the corresponding series to special values of the Dirichlet $L$-functions. For example, we establish the following identities conjectured by Sun:\[\sum_{k=0}^\infty\binom{2k}{k}^3\left[ \mathsf H_{2k}^{(2)}-\frac{25}{92}\mathsf H_{ k}^{(2)} +\frac{735L_{-7}(2)-86π^{2}}{1104}\right]\frac{1}{4096^{k}}=0,\]\[\sum_{k=0}^\infty\binom{2k}k^3\left[\mathsf H_{2k}^{(3)}-\frac{43}{352}\mathsf H_k^{(3)}\right]\frac{42k+5}{4096^k}=\frac{555ζ(3)}{77π}-\frac{32G}{11},\] where $ \mathsf H^{(r)}_k:= \sum_{0<n\leq k}\frac{1}{n^r}$, $ L_{-7}(2):= \sum_{n=1}^\infty\left(\frac{-7}{n}\right)\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{4^{2}}-\frac{1}{5^{2}}-\frac{1}{6^{2}}+\frac{1}{8^{2}}+\cdots $, $ G:= \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^2}$, and $ ζ(3):= \sum_{n=1}^\infty\frac1{n^3}$.
We prove and generalize some recent conjectures of Z.-W. Sun on infinite series whose summands involve products of harmonic numbers and several binomial coefficients. We evaluate various classes of infinite sums in closed form by interpreting them as automorphic objects on the moduli spaces for Legendre curves $Y^{ g+1}=(1-X)^{ g}X(1-t X)$ of positive genera $ g\in\{1,2,3,5\}$.
By exploring the theory of Guillera-Rogers, we evaluate some infinite series whose summands are quadratic irrationals, in terms of $π$ and special values of Dirichlet $L$-functions $ L_d(2)\equiv L(2,(\frac d\cdot)):=\sum_{k=1}^\infty\left( \frac{d}{k} \right)\frac1{k^2}$. Applying Kronecker's theorem to linear combinations of lattice sums, we obtain geometrically convergent series for $ L_{-56}(2)$, $ L_{-68}(2)$, $ L_{-87}(2)$, $ L_{-111}(2)$, and $ L_{-116}(2)$, which go beyond the solvable cases of Guillera-Rogers.
Through the application of an evaluation technique based on cyclotomic multiple zeta values recently due to Au, we solve open problems on inverse binomial series that were included in a 2010 analysis textbook by Chen.
We introduce and prove evaluations for families of multiple elliptic integrals by solving special types of ordinary and partial differential equations. As an application, we obtain new expressions of Ramanujan-type series of level 4 and associated singular values for the complete elliptic integral $\mathbf K$ with integrals involving $\mathbf K$.
Using cyclotomic multiple zeta values of level $8$, we confirm and generalize several conjectural identities on infinite series with summands involving $\binom{2k}k8^k/(\binom{3k}k\binom{6k}{3k})$. For example, we prove that \[\sum_{k=0}^\infty\frac{(350k-17)\binom{2k}k8^k} {\binom{3k}k\binom{6k}{3k}}=15\sqrt2\,π+27\] and \[\sum_{k=1}^\infty\frac{\left\{(5k-1)\left[16\mathsf H_{2k-1}^{(2)}-3\mathsf H_{k-1}^{(2)}\right]-\frac{12(6k-1)}{(2k-1)^2}\right\}\binom{2k}k8^k} {k(2k-1)\binom{3k}k\binom{6k}{3k}}=\frac{π^3}{12\sqrt2},\] where $\mathsf H^{(2)}_m$ denotes the second-order harmonic number $\sum_{0<j\leq m}\frac1{j^2}$.
We perform polylogarithmic reductions for several classes of infinite sums motivated by Z.-W. Sun's related works in 2022--2023. For certain choices of parameters, these series can be expressed by cyclotomic multiple zeta values of levels $4$, $5$, $6$, $7$, $8$, $9$, $10$, and $12$. In particular, we obtain closed forms of the series $$\sum_{k=0}^\infty\frac{x_0^k}{(k+1)\binom{3k}k} \ \ \text{and}\ \ \sum_{k=1}^\infty\frac{x_0^k}{k^2\binom{3k}k}$$ for any $x_0\in(-27/4,27/4)$.
The hyper-Mahler measures $m_k( 1+x_1+x_2),k\in\mathbb Z_{>1}$ and $m_k( 1+x_1+x_2+x_3),k\in\mathbb Z_{>1}$ are evaluated in closed form via Goncharov-Deligne periods, namely $\mathbb Q$-linear combinations of multiple polylogarithms at cyclotomic points (complex-valued coordinates that are roots of unity). Some infinite series related to these hyper-Mahler measures are also explicitly represented as Goncharov-Deligne periods of levels $1$, $2$, $ 3$, $4$, $6$, $8$, $10$ and $12$.
We prove and generalize several recent conjectures of Z.-W. Sun surrounding binomial coefficients and harmonic numbers. We show that Sun's series and their analogs can be represented as cyclotomic multiple zeta values of levels $N\in\{4,8,12,16,24\}$, namely Goncharov's multiple polylogarithms evaluated at $N$-th roots of unity.
Brain-computer interfaces (BCIs), invasive or non-invasive, have projected unparalleled vision and promise for assisting patients in need to better their interaction with the surroundings. Inspired by the BCI-based rehabilitation technologies for nerve-system impairments and amputation, we propose an electromagnetic brain-computer-metasurface (EBCM) paradigm, regulated by human's cognition by brain signals directly and non-invasively. We experimentally show that our EBCM platform can translate human's mind from evoked potentials of P300-based electroencephalography to digital coding information in the electromagnetic domain non-invasively, which can be further processed and transported by an information metasurface in automated and wireless fashions. Directly wireless communications of the human minds are performed between two EBCM operators with accurate text transmissions. Moreover, several other proof-of-concept mind-control schemes are presented using the same EBCM platform, exhibiting flexibly-customized capabilities of information processing and synthesis like visual-beam scanning, wave modulations, and pattern encoding.
This article reports on some recent progresses in Bessel moments, which represent a class of Feynman diagrams in 2-dimensional quantum field theory. Many challenging mathematical problems on these Bessel moments have been formulated as a vast set of conjectures, by David Broadhurst and collaborators, who work at the intersection of high energy physics, number theory and algebraic geometry. We present the main ideas behind our verifications of several such conjectures, which revolve around linear and non-linear sum rules of Bessel moments, as well as relations between individual Feynman diagrams and critical values of modular $L$-functions.
Spectral analysis is performed on the Born equation, a strongly singular integral equation modeling the interactions between electromagnetic waves and arbitrarily shaped dielectric scatterers. Compact and Hilbert--Schmidt operator polynomials are constructed from the Green operator of electromagnetic scattering on scatterers with smooth boundaries. As a consequence, it is shown that the strongly singular Born equation has a discrete spectrum, and that the spectral series $ \sum_λ|λ|^2|1+2λ|^4$ is convergent, counting multiplicities of the eigenvalues $ λ$. This reveals a shape-independent optical resonance mode corresponding to a critical dielectric permittivity $ ε_r=-1$.
We carry out quantitative studies on the Green operator $ \hat{\mathscr G}$ associated with the Born equation, an integral equation that models electromagnetic scattering, building the strong stability of the evolution semigroup $\{\exp(-iτ\hat{\mathscr G})|τ\geq0\} $ on polynomial compactness and the Arendt-Batty-Lyubich-Vũ theorem. The strongly-stable evolution semigroup inspires our proposal of a non-perturbative method to solve the light scattering problem and improve the Born approximation.
We perform quantitative spectral analysis on the Born equation, an integral equation for electromagnetic scattering that descends from the Maxwell equations. We establish norm bounds for the Green operator associated with the Born equation, thereby providing numerical tools for error estimates of the Born approximation to light scattering problems.
Through algebraic manipulations on Wrońskian matrices whose entries are reducible to Bessel moments, we present a new analytic proof of the quadratic relations conjectured by Broadhurst and Roberts, along with some generalizations. In the Wrońskian framework, we reinterpret the de Rham intersection pairing through polynomial coefficients in Vanhove's differential operators, and compute the Betti intersection pairing via linear sum rules for on-shell and off-shell Feynman diagrams at threshold momenta. From the ideal generated by Broadhurst--Roberts quadratic relations, we derive new non-linear sum rules for on-shell Feynman diagrams, including an infinite family of determinant identities that are compatible with Deligne's conjectures for critical values of motivic $L$-functions.
Let $I_0$ and $K_0$ be modified Bessel functions of the zeroth order. We use Vanhove's differential operators for Feynman integrals to derive upper bounds for dimensions of the $\mathbb Q$-vector space spanned by certain sequences of Bessel moments \[ \left\{\left.\int_0^\infty [I_0(t)]^a[K_0(t)]^b t^{2k+1}\mathrm{d}\, t\right|k\in\mathbb Z_{\geq0}\right\},\]where $a$ and $b$ are fixed non-negative integers. For $ a\in\mathbb Z\cap[1,b)$, our upper bound for the $ \mathbb Q$-linear dimension is $\lfloor (a+b-1)/2\rfloor$, which improves the Borwein-Salvy bound $\lfloor (a+b+1)/2\rfloor$. Our new upper bound $\lfloor (a+b-1)/2\rfloor$ is not sharp for $ a=2,b=6$, due to an exceptional $ \mathbb Q$-linear relation $\int_0^\infty [I_0(t)]^2[K_0(t)]^6 t\mathrm{d}\, t=72\int_0^\infty [I_0(t)]^2[K_0(t)]^6 t^{3}\mathrm{d}\, t$, which is provable by integrating modular forms.