Papers

A recent manuscript of OpenAI asserts that Landau-Siegel zeros do not exist. Here, we give a detailed account of its argument, arranged as a direct proof with explicit constants. There are absolute constants $q_0$ and $\delta_0>0$ such that, for every primitive real Dirichlet character $\chi$ of conductor $q\ge q_0$, every real zero $\beta\in(0,1)$ of $L(s,\chi)$ satisfies $(1-\beta)\log q\ge\delta_0$, and one can take $q_0=10^{10}$ and $\delta_0=3.0\times 10^{-3}$. Combined with a recent computation of Lu, Zaman, and Zhao, this result yields the bound $(1-\beta)\log q\ge 3.0\times 10^{-3}$ for every conductor $q\ge 3$. The proof is an interpolation determinant argument in the biquadratic field generated by $\sqrt{2}$ over the quadratic field attached to $\chi$. A real zero close to one forces $\chi(p)=-1$ for most primes $p$, and the Frobenius congruence at these primes makes a certain nonzero determinant divisible by higher powers of these primes than its Archimedean size allows. The underlying ideas are those of the OpenAI manuscript. The explicit constants, and the presentation, are new.

We extend the classical trace Paley-Wiener theorem for $p$-adic reductive groups, given by Bernstein, Deligne and Kazhdan in 1986, to a relative case. That is when $E/F$ is a quadratic extension of $p$-adic fields, the main result is a scalar Paley-Wiener theorem for relative Bessel distributions on $\mathrm{GL}_n(F)\backslash \mathrm{GL}_n(E)$. These distributions are relative character of the form $$\pi\mapsto I_\pi (f), \ f\in C_c^\infty(\mathrm{GL}_n(E)),$$ as $\pi$ ranges over $\mathrm{GL}_n(F)$-distinguished irreducible tempered representations, and are constructed from a $\mathrm{GL}_n(F)$-invariant functional and a Whittaker functional. Using the local Langlands correspondence, and the base-change from a unitary group, we show that the relative characters can be described as elements of the "generic" Bernstein center of the unitary group $U(n)$.

Let $h\geq2$ be an integer and let $\mathbb{N}_0=W_1\cup\cdots\cup W_h$ be a partition into infinite sets. For $W\subseteq\mathbb{N}_0$, let $A(W)$ denote the set of all finite nonempty sums of distinct powers $2^w$ with $w\in W$. We give a necessary and sufficient condition for $A=A(W_1)\cup\cdots\cup A(W_h)$ to be a strongly minimal asymptotic basis of order $h$. This provides a complete answer to a problem posed by Nathanson in 1988.

Let $\zeta^t(\boldsymbol{k})=\sum_{d=1}^{r}S_d(\boldsymbol{k})t^{r-d}$ be the interpolated multiple zeta value of an admissible index $\boldsymbol{k}$ of depth $r$. We show the monotonicity property of coefficients $S_1(\boldsymbol{k})\ge\cdots\ge S_r(\boldsymbol{k})>0$. In the proof, we represent the coefficients as integrals over a common measure space and compares the integrands pointwise. As consequences, we determine the sign and vanishing of $\zeta^{-1}(\boldsymbol{k})$ and locate the zeros of $\zeta^t(\boldsymbol{k})$ in the closed unit disk.

We establish lower bounds for the non-vanishing proportions of primitive cubic, quartic and sextic Dirichlet $L$-functions over $\F_q[T]$ in the non-Kummer setting. Adapting the truncated-exponential method of Gao and Zhao, we improve the cubic lower bound of David, Florea and Lalín and obtain positive lower bounds for quartic and sextic families in which every ramified local character has full order. For each family, the lower limit of the non-vanishing proportion exceeds $e^{-e^{66}}$ as the conductor degree tends to infinity with $q$ fixed. We also establish uniform twisted first moments in the quartic and sextic cases, yielding an asymptotic formula for the sextic first moment. All results are unconditional.

Let $V\subset\mathbb P^2\times\mathbb P^2$ be the threefold $x_0y_0^2+x_1y_1^2+x_2y_2^2=0$, and let $U$ be the complement of its coordinate hyperplane sections. For the anticanonical height $H([\mathbf x],[\mathbf y])=|\mathbf x|^2|\mathbf y|$, evaluated on primitive integral representatives, we prove \[ N_U(B)=\frac{12\pi-14}{\zeta(2)\zeta(3)}B\log B+o(B\log B). \] The two projections combine lattice counting with uniform conic estimates. The main arithmetic step is an average of the fiber singular series, whose two main terms are related by Hilbert reciprocity. A moment estimate controls the full bad-prime part, and shape estimates permit the errors to be summed over the height region. We also check the leading constant with Peyre's constant.

Recently, OpenAI announced a uniform $7/8$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$, together with its transfer to Dirichlet $L$-functions. Assuming the analytic inputs of that work, we improve the boundary from $7/8=0.875$ to $8748867/10000000=0.8748867$ for the same families. The improvement combines a relaxed direct estimate with an amplified first moment of the reflected numerator, obtained from the existing fourth-moment estimate. A two-stage parameter argument yields the stated boundary without strengthening the inverse-moment estimate.

Let $\Lambda_m$ be the Barnes-Wall lattice of dimension $2^m$, normalized to have minimum squared norm $d_m:=2^{m-1}$. We determine its complete norm spectrum strictly between $d_m$ and $2d_m$, proving that, for every integer $m\ge1$, $\{\|x\|^2:x\in\Lambda_m,\ d_m<\|x\|^2<2d_m\}=\{2d_m-2^j:j\in\mathbb Z,\ \lceil m/2\rceil\le j\le m-2\}$. In particular, the norm spectrum in this interval is nonempty for $m\ge4$. The proof uses a Gaussian-integer model in which lattice automorphisms arising from Clifford unitaries transform a vector into a pair of coordinate subvectors in a lower-dimensional lattice. Divisibility of the differences of their squared norms and a matrix Parseval identity provide the arithmetic and averaging constraints needed for induction on $m$. Every listed norm is realized explicitly by a sum of two minimal vectors of the lattice.

For a prime $p\ge7$, let $S(C_p\times C_p)$ be the set of integer group determinants and $e(p)$ the least exponent for which $p^{e(p)}\mathbf Z\subseteq S(C_p\times C_p)$. Let $r_p$ be the least integer $r\ge1$ satisfying $(2r+1)i>r(p-1)$ for every irregular pair $(p,2i)$ with $r\mid i$. We prove the unconditional inclusion $p^{p+3+2r_p}\mathbf Z\subseteq S(C_p\times C_p)$. Equality $e(p)=p+3+2r_p$ holds when Vandiver's conjecture holds for $p$ and $r_p\le(p-5)/2$, hence for every prime $7\le p<2^{31}$. The first exception to $e(p)=p+5$ is $p=103$, where $e(103)=116$. A polynomial obstruction from Bernoulli numbers supplies the lower bound, and circular units give the construction. For regular primes, we characterize the entire determinant set through explicit affine systems over $\mathbf F_p$ and principality conditions in $\mathbf Z[\zeta_p]$. For regular $p\ge11$, prime cofactors cannot occur at valuation $p+3$; at valuation $p+4$ they have natural density $(p+1)/(hp^{(p-3)/2})$, where $h$ is the class number of $\mathbf Q(\zeta_p)$. We also show that $p^{p+5}$ is the least absolute value of a nonzero $p$-divisible determinant for these regular primes.

We prove that, for any number field K and and d at least 2, there are at most finitely many monic, centred, post-critically finite polynomials of degree d with coefficients in the extension of K by all roots of unity.

We prove two supercongruence conjectures of Li–Long–Tu concerning truncated very-well-poised ${}_7F_6(1)$ series and Fourier coefficients of modular forms. The proofs are based on a common deformation method. By deforming the original truncated hypergeometric series to suitable terminating series, we are able to apply classical hypergeometric transformations together with Mishev's $W(D_5)$ symmetry. This reduces both problems to congruences for lower-rank hypergeometric sums that can be treated using the $p$-adic methods of Long–Tu–Yui–Zudilin. Although the two cases require different local arithmetic calculations, their global proof structure is the same. The resulting argument gives a unified explanation of these two supercongruences and suggests that the same deformation-and-symmetry method may apply more broadly.

In a previous work, we defined a quotient graph of the Bruhat-Tits tree associated to the degree two special linear group over a local field whose vertex set is in correspondence with the maximal orders in a certain number of spinor genera. This graph has been applied to the study of the structure of some arithmetic groups, their presentations and their (co)homology groups. The genera of Eichler orders, which, for instance, are related to the problem of finding a basis that simultaneously splits two vector bundles with the same generic fiber, can often be described in terms of simpler genera via quotients. Furthermore, some embedding problems of orders can be studied via quotient graphs. These results provide a geometric framework for the study of maximal orders over function fields. In this article, we extend that construction to a higher dimensional setting through the use of singular chain complexes. Then we describe the set of vertices in this quotient that correspond to split maximal orders and the topology of the corresponding region of the quotient complex. More precisely, we show that these “split vertices” fall into a finite number of quotients of apartments by suitable subgroups of the corresponding Weyl group, the symmetric group. We relate these “quotient apartments” with the cuspidal sectors defined by previous authors who studied the quotient by the group of integral points of the special linear group. Both quotients are indeed related by a ramified cover that is unramified on cusps, as proved here.

In this paper, it is proved that every sufficiently large odd integer $n$ can be represented as \begin{equation*} n=p_1^2+p_2^2+p_3^4+p_4^4+p_5^4+p_6^5+p_7^7, \end{equation*} where $p_1,\dots,p_7$ are prime variables. This result constitutes an enhancement upon a previous result of Andrade [1].

Denote by $\Pi_\gamma(x)$ the number of primes $p \leqslant x$ such that $[p^{1/\gamma}]$ is also a prime. In this paper, it is proved that the expected asymptotic formula \begin{equation*} \Pi_\gamma(x)=\frac{\gamma x}{(\log x)^2} (1+o(1)) \end{equation*} holds for $\gamma > 15/13$, which constitutes an improvement upon the previous result of Balog [2] who showed that the above asymptotic formula holds for $\gamma > 6/5$.

The Fibonacci, Lucas, and Pell sequences satisfy certain families of “polynomial” identities. While a few members of these families are well-known, we are not able to find a general statement of these identities in the mathematical literature. We show that such polynomial identities exist for many generalized Fibonacci sequences, and we classify sequences that have these identities obtained by two distinct approaches. One approach uses the Girard-Waring formula, while the other relies on Binet's formula and linear algebra.

We prove Kitaoka's conjecture for all real cyclotomic fields, i.e. for fields of the form $\mathbb{Q}(\zeta_k+\zeta_k^{-1})$ where $\zeta_k$ is a primitive $k$th root of unity. More precisely, we show that such a field admits a universal classical ternary quadratic form if and only if it is a real quadratic field.

Let $K>1$ be a real number and let $A\subsetneq \mathbb Z$ be a finite and nonempty subset. Suppose $|A+A|<K|A|$. A fundamental open problem is to characterize the size of higher iterated sumsets $nA = A+\cdots+A$ ($n$-fold sumset). In 2004, Terry Tao formally posed the problem of determining the optimal growth function governing iterated sumsets of small-doubling sets. In this manuscript, we give a solution to this long-standing open problem for large enough $n$.

In this paper, we prove two conjectures concerning parametric $q$- supercongruences related to Van Hamme's $\mathrm{(G.2)}$ supercongruence. First, we confirm Tang\rq s conjecture \cite[Conjecture $4.1$]{Tang G2} by establishing the corresponding congruences modulo $[m]\Phi_m(q)^2$. Second, we prove a conjecture of Wang and Xu \cite[Conjecture $2$]{WangXu24}, extending their $q$-supercongruence modulo $\Phi_m(q)^5$ to modulo $[m]\Phi_m(q)^4$ when $m\equiv1\pmod4$.

We prove that for every integer $n>2$ the largest good lattice point matrix $U_n=[u_h:h\in H_n]$, whose columns are generated by all integers $h<n$ coprime to $n$, has rank exactly $\varphi(n)/2+1$. The upper bound follows from the pairing identity $u_h+u_{n-h}=(n,\dots,n,2n)^{\top}$. For the lower bound the columns are centered and the resulting orbit under the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^{\times}$ is analysed by means of odd Dirichlet characters; the essential arithmetic input is the non-vanishing of $\sum_{a}a\,\psi(a)$ for odd primitive characters $\psi$, which is equivalent to Dirichlet's theorem $L(1,\psi)\neq 0$.

We study the Gibbs measure of $H(h) = \Re\sum_{p^{j} \leq X}U_{p}^{j}/(j p^{j(\sigma + i h)})$, a random field modeling large values of the log-modulus of the Riemann zeta function on and near the critical line, where the $U_{p}$ are independent and uniformly distributed on the complex unit circle. Writing $t = \log \log T$, the cutoff $X$ and real part $\sigma$ may vary with $T$ subject to $T^{1/t^{\Phi}} \leq X \leq T^{1/t^{\phi}}$ and $0 \leq \sigma - 1/2 \leq C_{\sigma}t^{\Phi}/\log T$, for fixed $\Phi \geq \phi > 1/2$ and $C_{\sigma} > 0$. On $I_{\theta} = [ - (\log T)^{\theta},(\log T)^{\theta}]$, define the Gibbs measure $G_{\beta,\sigma,T}^{\mathrm{H}}(\mathrm{d}h) := e^{\beta H(h)}\mathrm{d}h/\int_{I_{\theta}}e^{\beta H(h')}\mathrm{d}h'$. For every fixed $\theta > - 1$ and inverse temperature $\beta > 0$, the limiting joint distributions of correlations (overlaps) among points sampled from this measure are determined after averaging over the Haar phases. At and below $\beta_{c}(\theta) = 2\sqrt{1 + (\theta \vee 0)}$, the limiting two-overlap distribution is concentrated at $q_{0} = |\theta \wedge 0|$. Above this threshold, it assigns masses $\beta_{c}(\theta)/\beta$ and $1 - \beta_{c}(\theta)/\beta$ to $q_{0}$ and $1$, respectively, and the limiting joint overlap distribution is described by a Poisson-Dirichlet process of parameter $\beta_{c}(\theta)/\beta$. Approximate Ghirlanda-Guerra identities are shown to hold uniformly for bounded deterministic functions of the sampled positions. This paper is the first in a series devoted to extending these results to $\log|\zeta|$ near the critical line.

Let $F$ be a $p$-adic field, let $k,c\ge1$, and let $\rho$ be a $(k,c)$ representation of $\mathrm{GL}_{kc}(F)$: a complex representation of finite length whose wave front set has $(k^c)$ as its unique maximal orbit, and which admits a $(k,c)$ functional that is unique up to scaling. We prove that the functions in the $(k,c)$ model of $\rho$ are of moderate growth, by relating this model to the Zelevinsky model of $\rho$, and obtain as a consequence a bound for the restriction of these functions to the standard Levi subgroup associated with the composition $(c^k)$ of $kc$. As applications, we study convergence problems of several integrals involving $(k,c)$ models: integrals from the generalized doubling method, and a new family of integrals which extends the Rankin–Selberg integrals for Speh representations of Lapid and Mao.

In 2010, M. A. Chaudhry et al. introduced a class of functions Sigma satisfying an L^1-condition. They proved that every function phi in Sigma gives rise to the functional equation for the Riemann zeta function. They also provided an example of a function outside Sigma for which the functional equation still holds. In this paper, we present several examples of Bessel functions of order zero that do not belong to Sigma but nevertheless give rise to the functional equation for the Riemann zeta function. In particular, we show that J_0(a sqrt(x)), J_0(ax), and Y_0(x) exhibit this phenomenon.

In this paper, by normalizing the trace-zero norm forms according to the parity of orders of definite rational quaternion algebras, we give an explicit formulation to the even Clifford correspondence followed by the reciprocity of ternary forms. We obtain an exact identity between ternary representation numbers and the number of elements of an order with prescribed reduced norm and trace, which applies to all orders without squarefreeness constraint on the reduced discriminant. Applications include a criterion for realizing ternary forms as trace-zero norm forms, a formula for traces of Brandt matrices in terms of representation numbers, and explicit theta series relations associated with local reciprocity. We also describe reciprocal orders and order chains, and interpret the ternary forms occurring in the congruent number problem through their associated quaternion orders.

We solve an open problem posed by Richard Guy in his book Unsolved Problems in Number Theory. We prove that, for every integer $n\geq10$, there exist infinitely many $n\times n$ integer matrices $A$ such that every entry is $\ge2$ and $\det A=\det A^{[3]}=1$, where $A^{[3]}$ denotes the entrywise cube of $A$. The matrices may be chosen compatibly, so they can form the leading principal blocks of a single infinite positive integer array. The order-ten case is reduced to an inhomogeneous system consisting of one linear and one diagonal cubic equation. The circle method then provides infinitely many positive integral solutions. To pass from order $n$ to order $n+1$, we introduce a bordering construction that preserves the signed cofactor vectors of both the matrix and its entrywise cube. Iterating this construction yields the result in every order $n\geq10$.

Let $T$ be the Gauss map. For each $\alpha\in [0,1]$, let $E(\alpha)$ be the survivor set consisting of all $x\in [0,1]$ whose orbit $\{T^k(x):k\ge 0\}$ never enters the interval $(\alpha,1]$. We prove that the set-valued function $\alpha\mapsto E(\alpha)$ is increasing and locally constant almost everywhere. Moreover, we explicitly characterize all intervals where the function $\alpha\mapsto E(\alpha)$ remains constant. We also describe the points at which the map $\alpha\mapsto E(\alpha)$ is not locally constant. Further, we obtain the Hausdorff dimension of $E(\alpha)$ for all $\alpha\in [0,1]$. We prove that the function $\alpha\mapsto \dim_H(E(\alpha))$ forms a devil's staircase on $(0,1]$, but is not continuous at $0$. This discontinuity implies that the result of Raith [Acta Math. Univ. Comenian. 63 (1994), 39-53] fails for piecewise monotone interval maps with infinitely many branches. Finally, we give an application quantifying the set of points sharing identical $\alpha$-continued fraction expansion and regular continued fraction expansion.

We develop a finite-tree formalism for sharp inequalities on real-valued, mean-centered $\operatorname{L}^1$-BMO over homogeneous $q$-ary filtrations. A reduction theorem identifies local and global sharp constants with the supremum of finite-depth constants. Finite anchoring gives compact optimization problems and rational extremizers under rational piecewise-linear hypotheses, while stable centering shows that imposing zero root average preserves the supremum over all depths. The principal application is the absolute-value map: we construct centered examples in which taking absolute values increases the BMO seminorm and obtain bounds strictly below the universal factor $2$. Jensen-defect Carleson functionals and martingale-difference envelopes provide further applications, illustrating distinct anchoring mechanisms and the differences between finite-depth rationality, stabilization, and global attainment.

The Ramanujan bound states that the degree of the trace of the $\mathfrak{p}$-Hecke operator on a space of weight $k$ Drinfeld cusp forms is at most $d(k-2)/2$, where $\mathfrak{p}$ is a prime of degree $d$ in $\mathbb{F}_q[T]$. We prove the strong Ramanujan bound, which states that at level 1, this bound can be improved to $d(k-(q+1))/2$. In addition, we classify when the strong Ramanujan bound is sharp; surprisingly, this turns out to depend on $\mathfrak{p}$ only through the parity of its degree. We also prove results related to the smallest slope of the Atkin operator and mod $p$ distributions of Frobenius polynomials of Drinfeld modules over finite fields.

We generalize Christol's theorem—which characterizes algebraic power series over a finite field in terms of automatic coefficient sequences—to completions of domains of finite type over a finite field~$\mathbb{F}_{q}$. Let $A$ be such a domain of dimension~$d$ and $\mathfrak{a}$ an ideal of depth~$d$. In the $\mathfrak{a}$-adic completion $\hat{A}$, choose an $\hat{A}$-regular sequence $\boldsymbol{x}=\{x_{1},\dots,x_{d}\}\subset\hat{\mathfrak{a}}$ and a set of representatives $E$ of $\hat{A}/(\boldsymbol{x})$ that is an $\mathbb{F}_{q}$-vector subspace consisting of algebraic elements. Every element $y\in\hat{A}$ then admits a unique expansion $y=\sum_{u\in\mathbb{N}^{d}}a_{u}x^{u}$ with coefficients $a_{u}\in E$. We prove that $y$ is algebraic over $A$ if and only if the coefficient sequence $\{a_{u}\}_{u\in\mathbb{N}^{d}}$ is $q$-automatic. The proof relies on a careful study of henselizations, which are shown to coincide with the algebraic closures inside the completions under certain conditions. As an application, we obtain a complete characterization of the automaticity of $\beta$-expansions for formal Laurent series over $\mathbb{F}_{q}$, answering questions posed by Hbaib–Mkaouar and Scheicher–Sirvent.

Let $\kappa_p(n)$ denote the maximum kissing number of a lattice $\mathcal{L} \subset \mathbb{R}^n$. We prove that $\kappa_p(n) \geq 2^{\Omega(\sqrt{n})}$ for all $p\ge 1$ and that $\kappa_p(n) \geq \Omega(\alpha_p^n)$ for all $p > 2$, where $\alpha_p > 1$ is a constant depending only on $p$. In particular, our former result improves on the previously best known lower bound of $\kappa_2(n) \geq 2^{\Theta(\log^2 n)}$ for the $\ell_2$ kissing number, which has stood for over $60$ years. (In concurrent and independent work, Laarhoven and Kominers [arXiv, 2026] showed a similar lower bound on $\kappa_2(n)$ using similar techniques.) In fact, our lower bounds follow from efficient (randomized) constructions of lattices with large kissing number. Our results use a general construction of lattices with large kissing number. Specifically, we show a simple equivalence between such lattices and a new object that we call an exact locally dense lattice. Closely related objects appear throughout the literature on computational lattice problems. Our constructions of lattices with large kissing number then follow as corollaries by showing that constructions due to Bennett and Peikert [RANDOM, 2023] (using Craig lattices) and Aggarwal and Stephens-Davidowitz [STOC, 2018] (using sparsification of $\mathbb{Z}^n$) in fact give exact locally dense lattices, as well as a new construction of exact locally dense lattices built from rotations of the $m$-dimensional checkerboard lattice $D_m$.

Motivated by the role of piecewise linear functions in ReLU neural networks, we study systems of dilated functions in the Lebesgue spaces $L_q([0,1]^d)$, $q\in(1,\infty)$. For a sequence $\tau\in\ell_1$ we consider the operator $T_\tau g(x)=\sum_{k=1}^\infty \tau_k g(kx)$ and the function system $\mathcal Z^\tau$ obtained by applying it to the trigonometric system. We show that $\tau\mapsto T_\tau$ is an isomorphism of algebras from $\ell_1$ with the Dirichlet convolution as multiplication onto an algebra of operators and connect the inverse of $T_\tau$ to the inverse of $\tau$ with respect to the Dirichlet convolution. We give explicit conditions on $\tau$, under which the resulting system $\mathcal Z^\tau$ is a Schauder basis of $L_q([0,1]^d)$ and, for $q=2$, a Riesz basis for which we estimate the Riesz constants in terms of $\tau$. We characterise the Riesz bases of this type in terms of a sequence space of Dirichlet convolution multipliers, and we treat mixed systems of cosine and sine functions with different sequences. Finally, we apply these results to the system generated by two piecewise linear functions inspired by ReLU networks and improve the known lower Riesz constant to $2/3$, independently of the dimension.