Papers

We prove the $p$-Frobenius problems proposed as Conjectures 7.1 and 7.5 developed by T. Komatsu and R. Pandey (Bull. Korean Math. Soc. 2025;62:1397–1409.) for two families of consecutive odd integers. For integers $r,L,n\ge0$, the bounded restricted partition function $p_{\le r}^{(\le L)}(\le n)$ counts partitions of $n$ into at most $r$ parts, each at most $L$. Thus the bounded restricted partition functions $p_{\le 3}^{(\le a)}(\le s)$and $p_{\le 3}^{(\le a+1)}(\le s)$ play central roles in the proofs. Their generating functions are Gaussian polynomials, whose symmetry and unimodality provide a common tool for treating both families.

With the help of a $_6\phi_5$ summation formula and Guo and Zudilin's method, we shall establish some Dwork-type $q$-supercongruences in this paper. When $q\to1$, these $q$-supercongruences are able to engender the corresponding supercongruences. One of them may be stated as follows: for any prime $p\geq5$ and any positive integer $s$, \begin{align*} &\sum_{k=0}^{p^s-1}(6k-1)\frac{(-\frac{1}{3})_k^3}{(1)_k^3} \equiv 0\pmod{p^{3s}}. \end{align*}

Let $\mathbb{K}$ denote an algebraically closed field of arbitrary characteristic. In this paper we provide bounds for the size of a prime $\ell$ for which there are curves defined over $\mathbb{K}$ admitting automorphism groups of order $\ell^2$. In addition, we also give a classification of the families of curves attaining the upper bounds for $\ell$ in both tame and wild case. Finally, we present the full automorphism groups of the curves attaining the highest bounds.

We construct the multiplicative chaos measures emerging from the Dirichlet series of the twisted Steinhaus multiplicative function. Our work has three main features. First, our argument treats the full subcritical and critical phases simultaneously. Second, we show that the same multiplicative chaos measure arises from any approximation to the Dirichlet series in which the prime cutoff is sent to infinity and the critical line is approached from the right at arbitrary speeds. Such results are known as "universality" results, and we show this via a novel and quick partial summation trick. Third, we show that the multiplicative chaos measure is mutually absolutely continuous with a GMC measure in the same phase, coupled on the same probability space, almost surely. Our work unifies and extends results from a recent breakthrough trilogy of papers by Gorodetsky and Wong.

We prove unconditionally that at least two thirds of the nontrivial zeros of the Riemann zeta function, counted with multiplicity, are simple and lie on the critical line, and that at least five sixths are distinct; the previous unconditional records are $\frac{5}{12}$ and $0.6603$. With the Montgomery–Taylor window the constants become $0.6725$ and $0.8362$. The argument makes Montgomery's 1973 deduction unconditional: the Riemann hypothesis, classically needed to read the zero side as a positive sum over real ordinates, is replaced by a rank-trace inequality applied to a finite compression of Weil's Hermitian form, with Sylvester's law of inertia handling off-line pairs. The analytic inputs are those of Aryan and of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh. The results extend to primitive Dirichlet $L$-functions and are formally verified in Lean 4.

Let $y^2 = f(x,T)$ be a hyperelliptic curve of genus $g\geq 1$, defined over $\mathbb{Q}(T)$. We prove the existence of infinitely many imaginary hyperelliptic curves with a fixed genus $g$ having a certain rank for $5\leq r\leq 4g+2$, and a similar result for real hyperelliptic curves with a fixed genus $g$ having a certain rank for $6\leq r\leq 4g+4$. We begin by constructing such curves and prove the rank using two methods. First, we apply the generalized Nagao's conjecture, which relates the first moment and the rank of the Jacobian variety $J_\mathcal{X}(\mathbb{Q}(T))$, and that the conjecture holds for our curves, making the result unconditional. Furthermore, we explicitly construct rational points in the Mordell-Weil group and use Shioda-Tate to prove that the rank is equal to $r$.

In this paper, we extend the classical problem of studying the distribution of $k$-free integers in arithmetic progressions to the setting of arbitrary number fields. Using the language of ray class groups, we establish asymptotic formulas, together with error terms, for the number of $k$-free ideals of bounded norm lying in a given ray class. In particular, our results show that $k$-free ideals are equidistributed among ray classes. We also obtain improved error estimates in the cases of ideal class groups and narrow class groups by using sharper ideal counting asymptotics due to Landau. Our results recover the classical formulas of Gegenbauer and Cohen–Robinson over $\mathbb{Q}$ and extend previous work of Benkowski, Nymann, and Sittinger to the setting of ray class groups. We also present explicit computational examples that illustrate the asymptotic formulas and the equidistribution of $k$-free ideals among ray classes.

We classify universal divisor-profile separation for coprime polynomial pairs of arbitrary degree and for all pairs of degree at most two. For \(A\subset\N\) and \(m\in\Z\), let \(d_A(m)\) count the members of \(A\) dividing \(m\). For coprime nonzero \(F,G\in\Z[x]\), universally unbounded separation between \(d_A(F(n))\) and \(d_A(G(n))\) occurs if and only if one of \(F,G\) is intersective, that is, has a root modulo every positive integer. More generally, an intersective factor separated from finitely many polynomial opponents yields simultaneous one-sided dominance against all of them. For pairs with common irreducible factors, let \(U,V\) be the products of the factors occurring only on the two respective sides. Universal separation forces \(UV\) to have a root modulo almost every prime; equivalently, its Galois action has no derangement. We resolve the remaining finite \(p\)-adic boundary for all pairs of degree at most two: separation holds exactly when \(UV\) and at least one of \(F,G\) are intersective, and the criterion is unchanged by contents or factor multiplicities. The same criterion holds, in arbitrary degree, for three linear support factors with arbitrary positive multiplicities. The proofs combine uniform almost-prime values on root progressions with an adaptive local-routing argument.

Let p > 3 be a prime, and let S_p denote the geometric isomorphism classes of supersingular elliptic curves in characteristic p whose j-invariants lie in F_p. For each negative fundamental discriminant -D for which p is inert in Q(sqrt(-D)), let m_i(D), i in S_p, be the integral coefficients of the corresponding Gross vector. We prove that the vectors (m_i(D) mod 2)_{i in S_p} span F_2^{S_p}. The key step reduces the parity of the representation numbers of Gross's ternary lattices to representation by rank-two sublattices perpendicular to Frobenius. Using Ibukiyama's explicit maximal orders, the resulting primitive binary forms are identified with those occurring in the Xiao–Zhou–Deng–Qu parametrization of supersingular elliptic curves over F_p. Class field theory and Chebotarev's theorem then allow the individual supersingular coordinates to be isolated. As a consequence, if E/Q has prime conductor p and positive Mordell–Weil rank, then every coefficient of its Brandt eigenvector indexed by S_p is even, proving a conjecture of Kazalicki and Kohen. Thus an odd coefficient at a rational supersingular class is an algebraic certificate of rank 0. Combining this parity theorem with formulas of Mestre and Gross–Kudla, we also prove that the modular degree of every positive-rank elliptic curve of prime conductor and root number +1 is divisible by 4. Consequently, Watkins' conjecture holds for all such curves of rank 2.

Convolution identities and recursive formulas have long played a role in the theory of holomorphic modular forms and their applications. These have served both as striking formulas and as fundamentally useful tools for applications to combinatorics. Recently, there has been renewed interest in examples arising from non-holomorphic modular forms. These include the famous Hurwitz-Kronecker class number relations dating to 1885, and groundbreaking work of Imamoğlu, Raum, and Richter from 2014. Imamoğlu, Raum, and Richter developed a theory of holomorphic projection for products of (vector-valued) harmonic Maass forms and holomorphic modular forms to produce many such formulas. This was related shortly thereafter by Duncan, Griffin, and Ono to replicable-type functions in the sense of Conway and Norton, and such recursions played a key role in their proof of the Umbral Moonshine Conjecture. Here, we develop new results on holomorphic projections of such functions which is more convenient for many natural cases. Imamoğlu, Raum, and Richter's choice corresponds to the case in which the generalized Pell equation $m^2 - Dn^2 = N$ has a square value for $D$, which has only finitely many solutions for a given value of $N$. Our main results cover the cases of general $D$, in particular those for which the Pell equation has infinitely many solutions. As an application, we resolve a recent conjecture of the second author. More generally, our approach yields 18 convolution identities for mock theta functions, which we boil down to 5 identities that can directly be used to prove the others. In the appendices, we also show how these identities can be proven by more direct $q$-series methods; however, the key utility of the holomorphic projection formulas is that they give a tool to automatically discover and verify such formulas.

We establish a criterion for the existence of a local Minkowski unit at $p$ that applies to all Galois extensions with Galois group isomorphic to the direct product of a non-$p$-group and a cyclic $p$-group. As applications, we construct non-abelian extensions admitting a local Minkowski unit at $p$ under various ramification conditions and analyze the Iwasawa module structure of units in $\mathbb{Z}_p$-extensions of number fields. We also extend our study to the case where the group of $p$-power roots of unity is nontrivial.

In this paper, we consider the asymptotic density of $\#\{p\leq x:P^+(p-1)\geq p^c\}$ and $\#\{n\leq x:P^+(n)<P^+(n+1)\}$, where $P^+(n)$ denote the largest prime factor of $n$. We show that for $x\rightarrow\infty$, one has \begin{align*} \mathop{\lim\sup}_{x\rightarrow\infty} \frac{1}{\pi(x)}\#\{p\leq x:P^+(p-1)\geq p^c\}\leq\frac{16}{5}\log\frac{1}{c} \end{align*} where $e^{-5/16}<c<1$, and \begin{align*} \#\{n\leq x:P^+(n)<P^+(n+1)\}>0.299x. \end{align*} The first result constitutes an improvement upon that of Ding and Wang (2025), who obatined $\mathop{\lim\sup}_{x\rightarrow\infty} \frac{1}{\pi(x)}\#\{p\leq x:P^+(p-1)\geq p^c\}\leq \frac{7}{2}\log\frac{1}{c}$. The second result improves a previous result $0.280$ by the author (2026). The key to the proof is that for a special class of convolution forms equipped with well-factorable weights, we may use the level \(x^{5/8-o(1)}\) for Pascadi's prime-distribution result with triple-well-factorable weights. We also use Pascadi's estimation of incomplete Kloosterman sums.

We consider Goldbach type sums corresponding to the Liouville function and prove the existence of sufficient cancellations. We also consider applications to sign patterns in the Liouville function.

The spectral side of the Kuznetsov trace formula for $GL(2)$ is governed by the Bessel-Kuznetsov integral transform $\check{\phi}(t)$. While classical bounds guarantee rapid decay of this transform for smooth, non-oscillatory test functions, modern applications in analytic number theory—particularly those involving twisted shifted convolution sums—frequently encounter test functions exhibiting a highly oscillatory linear phase $e(\alpha x)$. In this paper, we provide a rigorous and explicit asymptotic analysis of $\check{\phi}(t)$ in the semiclassical limit $t \to \infty$ under such oscillatory conditions. By applying the WKB approximation to the imaginary-order Bessel kernel, we identify a sharp phase transition dependent on the twist parameter $\alpha$. We prove that in the sub-critical regime ($\alpha \le 1/2\pi$), the transform decays rapidly. Conversely, in the super-critical regime ($\alpha > 1/2\pi$), the geometric oscillations resonate with the spectral kernel, yielding a localized main term of order $O(t^{-1})$ with a remarkably simplified arithmetic phase.

For each $r\in\{4,6\}$, we construct an explicit one-parameter family of elliptic curves over $\mathbb{Q}(i)$ containing infinitely many pairwise nonisomorphic curves genuinely defined over $\mathbb{Q}(i)$ with $j$-invariant $1728$ and rank exactly $r$. We construct explicit $\mathbb{Q}(i)$-rational points to bound the ranks from below. Kai's theorem on prime values of linear patterns over number fields provides specializations with controlled local behavior, allowing us to obtain matching upper bounds via $[1+i]$-descent. The construction extends the strategy of the author's earlier rank-$2$ paper by replacing a symmetric Gaussian-prime configuration with systems of binary linear forms satisfying several complementary square identities. In the rank-$6$ case, the support vectors attached to the three constructed points and $(0,0)$ span the self-dual Reed-Muller code $\mathrm{RM}(1,3)$, which also occurs as the kernel of the quadratic-residue Laplacian governing the Selmer group.

The objective of this work is to determine optimal local restrictions on the coefficients of integer and Gaussian integer continued fractions that imply convergence. We identify all minimal restrictions involving words of length two in the integer case, and we identify all reversible minimal restrictions of length two in the Gaussian integer case. In the integer setting, our classification is equivalent to a classification of minimal unavoidable words of length two in Conway–Coxeter quiddity sequences. We also construct a canonical set of restrictions of infinite cardinality that is strictly stronger than every finite set of restrictions.

In this paper, we first extend the Christoffel-Darboux formula for orthogonal polynomials to general three-term recurrence sequences, and then investigate the identities and congruences for $g_n(x)$ and $v_n(x)$ given by \begin{align*} &g_0(x)=1,\ g_1(x)=\frac{x+1}2,\ (n+1)^2g_{n+1}(x)=\Big(2n(n+1)+\frac{x+1}2\Big)g_n(x)-n^2g_{n-1}(x)\ (n\ge 1), \&v_0(x)=1,\ v_1(x)=x,\ (n+1)^3v_{n+1}(x)=(2n+1)(n(n+1)+x)v_n(x)-n^3v_{n-1}(x)\ (n\ge 1).\end{align*}

In this paper, we establish the $\mathcal{B}^4$-almost periodicity in the sense of Besicovitch for a suitably normalized error term associated with a broad class of arithmetical functions introduced by Chandrasekharan and Narasimhan. This result significantly strengthens $\mathcal{B}^2$-almost periodicity previously investigated in the literature for related error terms. By deriving truncated Voronoï-type formulas, we demonstrate that the normalized error term lies in the Besicovitch space $B^4$. As a consequence, we deduce that the error term admits a limit probability distribution and establish an explicit formula for its mean fourth power moment.

Let $f$ be a non-CM newform of weight $k\geq4$, level $N$ and trivial Nebentypus. Let $p\nmid N$ be an odd prime number that is ordinary for $f$ and denote by $\boldsymbol f^{(p)}$ the $p$-adic Hida family passing through $f$. Assuming very specific instances of two general conjectures in arithmetic algebraic geometry (injectivity of $p$-adic Abel-Jacobi maps, positive definiteness of archimedean height pairings à la Gillet-Soulé), we prove that, for all but finitely many $p$ as above, if the analytic rank of $f$ is $1$, then all but finitely many specializations of $\boldsymbol f^{(p)}$ of even weight and trivial Nebentypus have analytic rank $1$. This result provides evidence for Greenberg's "minimality conjecture" on analytic ranks in families of modular forms.

We record several elementary relations between spectral and isoperimetric parameters of a flat torus $\mathbb{T}_\Lambda = \mathbb{R}^n/\Lambda$ and the covariance structure of a fundamental domain $K$ for the lattice $\Lambda \subset \mathbb{R}^n$. For every measurable fundamental domain $K$ and nonzero vector $\xi$ in the dual lattice $\Lambda^*$, we observe the sharp directional variance estimate \[ \left\langle \operatorname{Cov}_K \xi,\xi\right\rangle \geq \frac{1}{12}. \] This yields a lower bound on the torus spectral gap $\lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda)$ (equivalently, the length of the shortest nonzero dual vector $\lambda_1(\Lambda^*)$) in terms of the maximal covariance of $K$: \[ \lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda) = 4\pi^2 \lambda_1(\Lambda^*)^2 \geq \frac{\pi^2}{3\left\|\operatorname{Cov}_K\right\|_{\mathrm{op}}}. \] Analogous sharp results are obtained for the isoperimetric profile and the Cheeger constant $D_{\mathrm{Che}}(\mathbb{T}_\Lambda)$ using an old argument of Hadwiger. In particular, when the Voronoi cell $K_\Lambda$ of a lattice with $\det \Lambda = 1$ is isotropic, the recent resolution of the Slicing Problem by Klartag and Lehec implies that \[ D_{\mathrm{Che}}(\mathbb{T}_\Lambda),\quad \lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda),\quad \lambda_1(\Lambda^*) \geq c > 0, \] where $c > 0$ is a universal constant independent of dimension $n$; this may be thought of as a positive resolution of the Kannan–Lovász–Simonovits conjecture for all flat tori. While there are lattices $\Lambda$ and corresponding Voronoi cells $K = K_\Lambda$ for which no dimension-independent converse inequality to the spectral-gap bound above can hold, we show that under a certain sectional tiling hypothesis, this inequality is in fact an equivalence (up to numerical constants).

The paper introduces novel classes of operators and investigates their applications to the study of special numbers. We utilize these operators to derive explicit expressions for degenerate Euler numbers and type 2 degenerate Euler numbers. Furthermore, we establish a relationship between degenerate Bernoulli numbers and degenerate Euler numbers. We note here that these operators emerge naturally when we study such explicit expressions and such a relationship

Let $f(t)\in\mathbb{Z}[t]$ be a nonconstant polynomial with nonzero discriminant and let $k\ge2$ be an integer. Inspired by the work of Alon and Bourgain, for sufficiently large prime $p\equiv1\pmod{k}$, we study cyclic orderings of subsets $A\subseteq \mathbb{F}_p$ for which $ f(a_i+a_{i+1})$ is a nonzero $k$-th power for every consecutive pair. By combining mixed character-sum estimates, Fourier analysis on $\mathbb{F}_p$, and spectral graph methods, we establish a threshold $c(p,k,f)$ such that every subset $A$ with $\#A\ge c(p,k,f)$ admits such a cyclic ordering. We also give lower and upper bounds for the optimal threshold.

In the present article, we consider two families of integer sequences, the $(k,r)$-Lorenz-Fibonacci sequences of the first and second kind, whose characteristic polynomial is $$p_{k,r}(x)=x^k-x^{k-1}-\cdots- x^r+x^{r-1}+\cdots +x+1,$$ where $k\geq 2r+1$, and $k\geq 3$. These families include the well-known $k$-bonacci sequence, when $r=0$. These sequences arise naturally in the study of the dynamics of the Lorenz attractor. We introduce two families of substitutions (in an alphabet of $k$ symbols) so that they are associated with each of the families of integer sequences and share the same polynomial. We study the main combinatorial properties of these substitutions.

We prove that the archimedean Asai $L$-factor attached to an irreducible generic representation of $\operatorname{GL}_n(\mathbb{C})$ can be expressed as a finite sum of Flicker local zeta integrals. For an archimedean local field $F$, we also prove that the exterior-square $L$-factor attached to an irreducible generic representation of $\operatorname{GL}_m(F)$ can be expressed as a finite sum of Jacquet–Shalika local zeta integrals.

Enrique González-Jiménez and Álvaro Lozano-Robledo proved that there exists a uniform bound on the levels $n$ for which the $n$-division field $\mathbb{Q}\left(E\left[n\right]\right) / \mathbb{Q}$ of an elliptic curve $E/\mathbb{Q}$ defined over $\mathbb{Q}$ is abelian. Assuming GRH (Generalized Riemann Hypothesis), Allen and Genao partially generalized this result by restricting to prime levels while allowing arbitrary number fields without rationally defined CM. In this paper, we remove the GRH assumption and prove that the property established by Allen and Genao is in fact equivalent to the absence of rationally defined CM.

Motivated by sequential space-filling designs for computer experiments, we study algebraic constructions of extensible point sets in the $d$-dimensional unit cube whose two-dimensional coordinate projections are all quasi-uniform. Our two constructions share a common parametrization in terms of finite configurations of distinct rational directions on the projective line $\mathbb{P}^1(\mathbb{Q})$. First, using a cubic number field, we construct explicit Kronecker sequences for which the mesh ratios of all two-dimensional coordinate projections remain uniformly bounded over every initial segment of length $N\ge 2$. Second, using a real quadratic field, a split prime, and a compatible $p$-adic embedding, we construct nested rank-1 lattice designs with the same uniform projection property at every nesting level. The proofs combine algebraic norm estimates with transference principles between simultaneous and dual Diophantine approximation, yielding lower bounds for the separation radii and upper bounds for the covering radii, both of optimal order in the number of points, uniformly over all coordinate pairs. We also investigate how the choice of rational projective coefficients affects the resulting mesh ratios. This leads to a minimax problem for finite configurations on $\mathbb{P}^1(\mathbb{Q})$, in which one seeks to minimize the maximum mesh ratio over all two-dimensional coordinate projections. These constructions provide extensible point sets with uniformly controlled bivariate geometry.

The Casas–Alvero conjecture asserts that a monic polynomial $f$ of degree $d$ over a field of characteristic zero sharing a non-constant factor with each of $f',\dots,f^{(d-1)}$ is the $d$-th power of a linear polynomial. Two reductions are carried out. Once a root is translated to the origin, the condition at index $i$ is vacuous unless the coefficient $a_{d-i}$ is nonzero, so the Casas–Alvero locus stratifies by the support of the centred normal form; prescribing in addition which root realises each surviving condition makes the system unit lower triangular and eliminates the coefficients of $f$ entirely. When one root suffices, what remains is a single integer, and that integer is MacMahon's determinant: the number of permutations of $\{1,\dots,d\}$ with descent set the support of $f$. Since every subset is a descent set, no counterexample has fewer than three recycled roots; in characteristic $p$ a prime dividing one of these counts but none of the accompanying integers $z_{i}$ is a bad prime, the largest count in degree $d$ being the Euler zigzag number $A_{d}$, so that the irregular prime $691$ is bad for $d=11$. The stratification also yields short proofs of the two characteristic-$p$ propositions on which all known cases rest; supports of two elements are settled separately in every characteristic, whence a counterexample has at least four terms in centred normal form and a further Gröbner-free criterion for bad primes.

This work proposes a semantic environment for arithmetic geometry in which passage between presentations carries an explicit weight. Morphism costs induce a Lawvere generalized distance by taking infima over transports, and heights satisfy the corresponding transport bounds. A labeled quantitative $(2,1)$-category retains invertible $2$-morphisms as coherent comparisons between transports. Its vertical groupoids admit a transport interpretation inspired by the identity types of Voevodsky's C-systems. If a height factors through the underlying object, then a vertical comparison has equal endpoints, whatever its transport cost. We formulate an intensional Szpiro inequality for elliptic packages, where the discriminant height transports along a morphism with a controlled model defect, and we relate a bound of this form to the $abc$ conjecture. The defect remains visible between non-minimal and normalized presentations, while conductor complexity stays attached to the underlying curve. We then treat the Birch and Swinnerton-Dyer conjecture in its pre-modularity form, using strong point-count asymptotics with an explicit convergence requirement. The refined constant retains separately typed arithmetic factors. A finite differential correction makes the period independent of the chosen differential and links its normalization to model transport. Within the declared finite-descent theory, Tate–Shafarevich finiteness reduces to finitely many closed prime columns of Selmer shadows and a uniform cutoff. For $y^2=x^3-q^2x$ with $q\equiv3\pmod8$ prime, two-isogeny descent closes the two-primary column uniformly. The statements admit expression in $\mathsf{ACA}_0$ relative to certified descent and Mordell–Weil data, with certified Cauchy names for the period and regulator.

We prove that, for every constant $\rho>1$, the Euclidean shortest vector problem is NP-hard to approximate within any constant factor $\rho$ under a deterministic polynomial-time many-one reduction. This extends our previous deterministic NP-hardness result from $\rho<\sqrt 2$ to arbitrary constants and gives a deterministic version of Khot's randomized arbitrary-constant theorem.

The famous 15-Theorem and 290-Theorem fully characterise universal quadratic forms over $\mathbb{Q}$. Similar theorems exist for every totally real number field, but only over $\mathbb{Q}(\sqrt5)$ the criterion set is explicitly known. We study criterion sets both theoretically and computationally: We develop the method of escalation over number fields, thus providing a simple proof of finiteness of the criteria and, more importantly, a practical tool for computing them. We illustrate this by explicit computations for $\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}(\sqrt3)$, obtaining a conjecture about the corresponding universality criteria; we also prove universality of many escalator lattices. Moreover, we develop the somewhat different theory of escalations for diagonal quadratic forms, obtaining the diagonal criterion set for $\mathbb{Q}(\sqrt5)$ and conjecturally for $\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}(\sqrt3)$.