Papers

When $k\ge 2$ and $s>0$, we establish that for each $\varepsilon>0$, one has \[ \int_{[0,1)^{k-1}}\biggl| \sum_{1\le x\le X}e(\alpha_2x^2+\ldots +\alpha_k x^k)\biggr|^{2s}\,{\rm d}\boldsymbol \alpha \ll X^{s+\varepsilon}+X^{2s-(k^2+k-2)/2}, \] confirming the main conjecture for a new family of exponential sums beyond those of Vinogradov (translation-dilation invariant) type. Our methods are based on very recent work of the author employing the nested efficient congruencing method. Consequently, analogues of our results hold also in the setting of number fields and function fields.

Let $F$ be a non-Archimedean local field of characteristic zero, let $D/F$ be the quaternion division algebra, and put $G_n = GL_n(D)$ and $H_n = Sp_n(D)$. We prove Prasad's revised conjecture on tempered representations with symplectic period. If $\rho$ is a unitary $H_r$-distinguished supercuspidal representation and $\delta_m(\rho)$ denotes the generalized Steinberg representation associated to $\rho$, set $B_0(\rho) = \rho, B_m(\rho) = \delta_{m+1}(\rho) \times \delta_m(\rho), m \geq 1.$ We prove that an irreducible tempered representation $\pi$ of $G_n$ is $H_n$-distinguished if and only if $\pi$ is isomorphic to the product over $i$ of $B_{m_i}(\rho_i),$ where each $\rho_i$ is a unitary distinguished supercuspidal representation.

We consider the one-level density of zeros of the Riemann zeta function weighted by its special values on the line ${\rm Re}(s)=\sigma$, where $\sigma$ is a fixed real number such that $1/2\le \sigma\le1$. We show that the density function for the Riemann zeta function changes if and only if $\sigma=1/2$.

In this paper, we prove that for any fixed base $b \ge 2$, there exists an infinite arithmetic progression of positive integers $k$ that are simultaneously generalized Sierpiński and generalized Riesel numbers in base $b$. More precisely, we construct such an arithmetic progression so that both $k\cdot b^n + 1$ and $k\cdot b^n - 1$ have at least two distinct prime factors for every positive integer $n$. Our approach uses covering systems, following the ideas of Erdős and later constructions of Harrington. The argument is then completed using Zsigmondy's theorem on primitive divisors together with properties of multiplicative orders.

We give a short and simple proof that an anisotropic generalized quadratic form over a quaternion division algebra with involution of the first kind, in arbitrary characteristic, remains anisotropic after scalar extension to any odd-degree field extension.

We prove weak approximation for Campana points on the projective line with log Fano boundary, and on projective $n$-space with either $n+3$ hyperplanes in general position or two transverse smooth quadrics, each of coefficient $1/2$.

The two-block odd partition function \(a(n)\) is the signed enumeration of partitions into exactly two distinct part sizes, each occurring an odd number of times. By separating the underlying representations according to the \(2\)-adic valuations of the two part sizes, we obtain a signed decomposition, completed by two classical theta-function evaluations, which explains the structural relation \(a(2^km)=a(m)+(2^{k-1}-1)\sigma(m)\), where \(m\) is odd, \(k\ge1\), and \(\sigma(m)\) is the sum-of-divisors function. Combined with a previously established arithmetic formula for \(a(n)\), this relation yields congruences along the dyadic progressions \(2^k(4n+3)\), as well as the families \(a\!\left(2^\alpha3^\beta(12n+11)\right)\equiv0\pmod3\) and \(a\!\left(2^\alpha(18n+15)\right)\equiv0\pmod3\). We also use a factorization of the generating function of \(a(n)\) involving \(\operatorname{pod}(n)\), the number of partitions in which odd parts are distinct and even parts are unrestricted, to establish a uniform reduction modulo every odd prime of the associated auxiliary series. The resulting theta-weighted sums involving \(\operatorname{pod}(n)\) are congruent to \(a(n)\) and therefore admit explicit evaluations involving the sum-of-divisors function and the character divisor sum associated with the nonprincipal Dirichlet character modulo \(4\). As special cases, we obtain infinite families of weighted congruences modulo \(3\), \(5\), and \(13\).

We prove the Baker–Rumely conjecture that the tau invariant of a metrized graph admits a positive lower bound proportional to its total length, with an absolute constant. We also construct simple cubic metrized graphs whose normalized tau invariants tend to $59/7260<1/108$, disproving the proposed universal constant $1/108$. The lower bound is independent of the genus, the number of edges, and the distribution of edge lengths. Its proof combines a second-moment inequality for Euclidean lattices with a partition of the edge coordinates of a cycle lattice into three independent sets. The counterexamples have only two edge lengths and admit an elementary resistance calculation. Through the tropical moment identity, the lower bound also gives a uniform estimate for the non-archimedean terms in height formulas for Jacobians.

For a positive integer $n$, let $$ S_1(n)=\{n\bmod k:1\leq k\leq\lfloor n/2\rfloor\}, \qquad s_1(n)=|S_1(n)|. $$ Baraskar and Vukusic proved $s_1(n)=\Xi n+O(n/(\log n\log\log n))$ for an explicit positive constant $\Xi$, and asked for a sharper error term. Their computations suggested that an $O(n^{1/3})$ bound might hold. We prove that, for every $\alpha>\sqrt2$, $$ \limsup_{n\to\infty} \frac{s_1(n)-\Xi n}{n\exp\{-\alpha\sqrt{\log n\log\log n}\}} =\infty, $$ and the corresponding limit inferior is $-\infty$. In particular, no fixed power saving is possible. We also prove the complementary upper estimate $$ s_1(n)=\Xi n+O\left( n\exp\{-\alpha\sqrt{\log n\log\log n}\} \right) $$ for every fixed $\alpha\in (0,\sqrt{2})$, sharpening the previous $O(n/(\log n\log\log n))$ bound. Thus, the constant $\sqrt2$ is the threshold for bounds of the form $n\exp\{-\alpha\sqrt{\log n\log\log n}\}$.

The Riemann hypothesis is equivalent to $F(x)$ belonging to the Laguerre–Pólya class. Brändén [J. Reine Angew. Math., 2011] proved that if an entire function in the Laguerre–Pólya class has nonnegative Taylor coefficients, then its coefficient sequence is infinitely log-concave. Consequently, the Riemann hypothesis implies the infinite log-concavity of $(\lambda_n)_{n\ge0}$. In this paper, we prove that the sequence $(\lambda_n)_{n\ge0}$ is strictly infinitely log-concave. This resolves a conjecture of Zhu [Math. Z., 2023]. The proof combines explicit complex-analytic estimates for the iterated logarithmic ratios, rigorous interval arithmetic for a finite range of indices, and a global closure argument.

This paper exposits relationships between the geometry of modular curves and the arithmetic of cyclotomic fields through the Eisenstein ideal. For an integer $N \ge 5$ and an odd prime $p$, we define two conjecturally inverse maps between the real part $P$ of the Eisenstein reduction of the homology $P$ of $X_1(N)$ and the real part of the $p$-part $Y$ of the second $K$-group of the $N$th cyclotomic integer ring. We recall a motivic construction of the map from $P$ to $Y$ and describe the recent proof of the Eisenstein property of the underlying map on modular symbols. We provide a fully equivariant construction of the second map from $Y$ to $P$ using the Eisenstein reduction of the first étale cohomology of $X_1(N)$. We then explain an equivariant construction of the Euler system of cyclotomic units using extension classes in relative cohomology groups of modular curves and describe its place in our program.

We prove several conjectures from the On-Line Encyclopedia of Integer Sequences about the lower and upper Wythoff sequences and the Fibonacci word. Among them are two of Kimberling's three conjectures on the number of ways to write $n=\lfloor h\varphi\rfloor+\lfloor k\varphi^2\rfloor$ with $h,k\ge1$ (A259598): exactly one way if and only if $n+1=2F$ for a Fibonacci number $F\ge2$, and exactly two ways if and only if $n+1\ge7$ is a Lucas number. The third conjecture, that no way exists if and only if $n+1$ is a Fibonacci number, was proved earlier by Kawsumarng et al. We also observe that Kimberling's conjecture on the gaps of the sums of two distinct terms of A003622 and of their complement (A333308, A333309) follows, after a shift by $2$, from earlier Walnut results of Shallit and of Bosma et al. on A260317, and we re-verify it. Next, we prove Kimberling's five 2025 conjectures on the gaps between positions where the Fibonacci word and the "Lucas word" take prescribed values (A383423-A383427). Finally, we prove a conjecture of Mathar on A285383, and we point out that a conjecture of Schmidt on A003250 follows from theorems of Carlitz, Scoville and Vaughan (1973); we also confirm it with Walnut. Most proofs are decision procedures run in the free prover Walnut, and we supply the complete command file.

We prove two supercongruence statements proposed by Z.-W. Sun for Domb numbers. The first concerns the unweighted truncated sum $$S_p=\sum_{n=0}^{p-1}D_n$$ and is governed by the two proper ideal classes of the order of discriminant $-60$ in $\mathbb{Q}(\sqrt{-15})$. We prove Sun's conjectured congruences modulo $p^2$ and obtain stronger formulas modulo $p^3$ in the split cases. The second concerns Domb sums weighted by the Lucas sequences associated with $X^2-11X+1$. Sun's original formulation has the exceptional counterexample $p=3$; after excluding this prime, we prove the corrected statement, including its modulo-$p^3$ refinement when both relevant quadratic characters are $+1$. Both results arise from one modular mechanism. The twisted Domb series is a modular period on $\Gamma_0(6)+\langle 3\rangle$ and is related by a quadratic pullback to Zagier's sporadic $C$-sequence. At primes inert in the imaginary quadratic field underlying the relevant complex multiplication, the truncated $C$-period is the Hasse invariant and vanishes at supersingular reduction; a finite pullback then gives divisibility modulo $p^2$. At split primes, explicitly constructed determinant-$1$ or determinant-$3$ matrices identify the Beukers unit root, yielding congruences modulo $p^3$. The two applications correspond respectively to orders of complex multiplication of discriminants $-60$ and $-156$.

We prove ten conjectures of Z.-W. Sun on Ramanujan-type series for $1/\pi^4$ and $\pi^4$ with harmonic numbers. They follow by comparing Taylor coefficients in new Borwein-Bradley-type identities, whose products depend only on $x^3$, $x^4$, $x^6$ or $x^8$. These identities come from parametric extensions of the series of Cullen and Zhao. We derive the extensions from K. C. Au's Wilf-Zeilberger (WZ) seeds, and their sums are trigonometric. Our proofs use Guillera's periodicity argument and give new proofs of the original series. In the same way, Dougall's $_5F_4$ sum gives half-integer analogues of the identities of Koecher, Borwein-Bradley and Cohen, whose Taylor coefficients are Apéry-like series for products of zeta values.

We study a one-parameter three-dimensional quadratic flow, denoted ETM-3D, on the benchmark domain $\rho\geq0$. Its homogeneous quadratic triad preserves Euclidean quadratic energy, while the full vector field has constant divergence -8. A shifted quadratic Lyapunov function yields an explicit positively invariant absorbing set and therefore global ultimate boundedness. The equilibria reduce generically to a scalar quartic, with a separate singular case at $\rho=\sqrt{5}$. Continuation reveals two saddle-node bifurcations, a generic transcritical bifurcation, a supercritical Hopf bifurcation, and a fold of limit cycles. Stable and unstable period-one branches meet at the fold with their nontrivial Floquet multipliers approaching +1. Just beyond this point, a 25-point logarithmic parameter study gives a laminar-length scaling exponent of -0.551 (95% confidence interval [-0.621,-0.481]), consistent with type-I intermittency. At larger $\rho$, a chaotic attractor coexists with an asymptotically stable equilibrium and periodic windows occur. Structural invariants are used to distinguish the flow from several classical quadratic chaotic families.

Klurman, Shparlinski and Teräväinen showed that there is a set of at most $(\log Q)^{1+\varepsilon}$ primes of polylogarithmic size whose subset products cover the unit group $(\mathbb{Z}/q\mathbb{Z})^\times$ for almost all moduli $q\le Q$. We show that $(1/\log2+o(1))\log Q$ primes suffice, which is optimal to first order, since $m$ primes have at most $2^m$ subset products. The proof combines a doubling argument of Erdős–Rényi type for random subset products in a finite abelian group, modified to tolerate a small set of exceptional characters, with a one-time “repair” of the few primitive characters whose $L$-functions have zeros near $s=1$. An exact weighted repair gives a variant with at most $(2/\log2+o(1))\log Q$ primes and an exceptional set described explicitly.

Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. We prove that \begin{align*} \sum_{n\leq x^{1/c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_cx^{1/c}+O_{\varepsilon,c}(x^{\theta_c+\varepsilon}), \end{align*} where $d_c$ is a suitable constant, $\theta_c<2/(3c+2)$ for $0<c<2/11$. This result constitutes an improvement upon that of Y. Feng (2024).

Let $C \subset \Pbb^2_k$ be a smooth projective plane curve of degree $d \ge 5$ over a number field $k$. We prove that any cubic arithmetic or geometric progression sequence on $C$ is finite. The proof rests on a prime-$p$ tower of cyclic Kummer covers controlled by the divisor of a linear-form quotient on $C$, combined with the Kadets–Vogt classification of curves with infinitely many cubic points, the Abramovich–Harris gonality descent, a Castelnuovo–Severi factorization for the elliptic maps supplied by Kadets–Vogt independently at consecutive levels of the tower, and a deck-transformation obstruction. We also obtain a conditional quartic result under a cyclotomic non-splitting hypothesis. The case $d = 4$ with $C(k) \neq \emptyset$ admits infinite cubic progressions, so no analogue of the main theorem can hold there.

For every prime $\ell\geq5$ and integer $n\geq2$, we construct an Ennola cubic field with a specified prime ideal above $\ell$ whose class has exact order $2^n$. For parameters satisfying two square conditions and explicit congruences, the same squareclass obstructions give four independent rational points on the associated elliptic curves. A Pell equation describes all such parameters. A genus-one base change gives five independent sections that generate a subgroup saturated at two. It also yields infinitely many elliptic curves over $\mathbb Q$, pairwise nonisomorphic over $\overline{\mathbb Q}$, with rank at least five.

We study iterated cotangent integrals with cyclotomic shifts and express their values along specified reference paths in terms of cyclotomic multiple zeta values. Differences of cotangent forms extend to a punctured projective line over a totally real cyclotomic field, giving a family of "real cyclotomic multiple zeta values." With rational tangential endpoints, these difference integrals are periods of mixed Tate motives over that field; integral endpoint data yield periods over its ring of integers after inverting primes dividing the level. The associated framed motivic classes generate a Hopf subalgebra under Goncharov's coproduct.

We represent constructive integrals in spatially cut-off $\phi^4_2$ by sequences of exponential-motivic periods. For a nonzero spatial cutoff, a single Galois transformation makes the partition periods diverge in modulus along every sequence of approximations satisfying the constructive estimates. For a fixed cutoff presentation, we characterize the subgroup of transformations admitting a bounded realization on the free field's $L^1$ space and prove that all fixed-observable limits on this subgroup are a common scalar multiple of their physical values. A one-variable quartic example shows that all transformed fixed-observable sequences can converge without such a bounded realization, with limits that are not proportional to the physical values.

We determine the connected geometric monodromy of the top-weight sheaves attached to linearly constrained Kloosterman families in every dimension n >= 4 and every characteristic p > max(2,n). The quantitative input is a fourth-moment identity whose auxiliary geometry depends on the moment degree rather than on the ambient family dimension. After an exact decoupling, the generic stratum is controlled by plane sections of the Cayley cubic, and a projective-ray argument gives the required power saving uniformly in each fixed dimension. On the geometric side, a local Fourier-transform calculation and Rojas-Leon's multiplicative-convolution formula produce a nonidentity unipotent boundary element. The fourth moment, reductivity, infinitude, and Larsen's alternative then give G_geom^0(W_n) = SL_r(n) without finite-group classification or numerical exceptional-case certificates. For comparison, we retain the independent Guralnick-Tiep route, including its sixth-moment and exact finite computations, but it is not used in the proof of the main theorem.

Let $n \geq 3$. We improve the large sieve inequality for $n$-th order Hecke characters over any number field containing the $n$-th roots of unity. Our bound is expected to be optimal in all ranges of parameters. Several applications are deduced, including a tight upper bound for the second moment of cubic characters ordered by conductor, asymptotics with lower order terms for the cubic second moment over squarefree moduli, and improved moment and zero density estimates for every $n \geq 3$.

In 1906, Maillet proved that every nonconstant rational function over $\mathbb{Q}$ maps Liouville numbers to Liouville numbers. Let $R$ be a nonconstant rational function with real algebraic coefficients, of degree $d$, and let $e$ be the degree of its minimal field of definition over $\mathbb{Q}$. We prove that, for every $m\geq 1$, the possible types of $R(\xi)$, as $\xi$ ranges over $U$-numbers of type $m$, are exactly the integers $r$ with $\lceil m/(ed)\rceil\leq r\leq em$. Moreover, every such type occurs for some $\xi$ of type $m$ in every nonempty open interval. Consequently, for $m\geq2$, the only rational functions with real algebraic coefficients preserving the class of $U$-numbers of type $m$ are the Möbius transformations over $\mathbb{Q}$.

Let $F$ be a number field and $f$ a normalized cuspidal Hecke eigenform on $\mathrm{GL}(2)$ over $F$, with $k/2$ as the central critical point of its $L$-function $L(f,s)$. Let $\mathfrak{p}$ be a totally split prime ideal of $F$ lying above an odd prime $p$ and coprime to the level ideal of $f$. In this article, we show that $L(f\otimes\phi, k/2)$ does not vanish for sufficiently ramified Hecke characters of $F$ of $p$-power order and $\mathfrak{p}$-power conductor. Our result improves previous works of Jaesung Kwon and Hae-Sang Sun, in which the additional hypothesis that $p$ does not divide the class number of $F$ is required.

One of the primary objectives of this paper is to construct generating functions for Apostol-Dedekind sums. These exponential generating functions greatly facilitate the discovery of new formulas and relations involving these sums. Other primary objectives of this paper are to define Apostol-Dedekind sums of higher-order and other certain family of finite sums, and to construct their generating functions. These generating functions include the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Array polynomials, the Stirling numbers, the Carlitz polynomials with their partial derivative equations, and the Hardy sums etc. Moreover, by using these functions, many novel formulas for the Apostol-Dedekind sums of higher-order are derived.

We prove the first quantitative bounds for the Gowers norms of the von Mangoldt function minus a Siegel corrected model function in short intervals. The result applies to intervals $(X,X+H]$ with $H\geq X^{5/8+\varepsilon}$ and gives quasipolynomial savings for the $U^k(X,X+H]$ norm. The main new ingredient is an efficient non-abelian Type II inverse theorem with explicit dependence on the nilsequence dimension. We prove this inverse theorem by applying Leng's efficient equidistribution theorem in four parameters directly to an unweighted corner correlation in four variables, hence replacing the factorisation and subgroup argument in the non-abelian Type II proof of Matomäki, Shao, Tao and the author. In joint work with Florian Richter, we use this quantitative result as one ingredient in proving the existence of infinitely many sum-product patterns of the form $\{x,x+y,xy\}$ in the shifted primes $\mathbb{P}-1$.

A shift space is said to have typical periodic optimization (TPO) if the set of Lipschitz functions whose unique maximizing measure is supported on a periodic orbit contains an open dense subset of the space of Lipschitz functions. We show that beta-shifts whose lexicographically largest point is a Sturmian sequence do not have TPO: on each such beta-shift there is a non-empty open set of Lipschitz functions, all of which have the Sturmian measure as their unique maximizing measure. These are the first known examples of beta-shifts without TPO, and, since they have the specification property, the first known examples of shift spaces with specification but without TPO. The corresponding beta-transformations do not have TPO in any space of Hölder functions on the interval. The main ingredient in the proof of these results is a rigidity property of Sturmian subshifts: modulo constants and Lipschitz coboundaries, the space of Lipschitz functions on such a subshift is one-dimensional.

We give two alternate formulations for the second moment formula of the Siegel transform over $\mathrm{Sp}(2n, \mathbb{Z}) \backslash \mathrm{Sp}(2n,\mathbb{R})$, originally established by Kelmer and Yu. We also provide several applications that demonstrate their usefulness.

We establish the Hausdorff dimension for exactly approximable matrices for all dimensions, including the inhomogeneous case. More precisely, let $\psi:\N\to(0,\infty)$ be a non-increasing approximation function and let $\tau_\psi$ denote its lower order. For $m,n\ge1$ and $\boldsymbol\theta\in\mathbb R^m$, denote $\Exact_{m,n}(\psi,\boldsymbol\theta)$ the set of $m\times n$ matrices that are $\psi$-approximable with shift $\boldsymbol\theta$, but are not $c\psi$-approximable for any $0<c<1$. It is shown that \[ \hdim \Exact_{m,n}(\psi,\boldsymbol\theta)=m(n-1)+\frac{m+n}{1+\tau_\psi},\ \ \ {\text{for}}\ \tau_\psi>\frac{n}{m}. \] We adopt a probabilistic viewpoint, i.e. by an averaged counting argument to quantify the distribution of rational vectors and subspaces $$\left\{\frac{\mathbf p+\boldsymbol\theta}{q}: (\mathbf p, q)\in \mathbb{Z}^{m+1}\right\},\ \ \ \bigg\{\big\{A\in \Mat_{m, n}(\R): A\mathbf q-\mathbf p-\boldsymbol\theta=0\big\}: (\mathbf p, \mathbf q)\in \mathbb{Z}^{m+n}\bigg\}.$$