50,133 papers in this slice of arXiv.
Adrian Barquero-Sanchez, Jack Heimrath, Bernd Sing +3
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
Zihan Zhang
We classify universal divisor-profile separation for coprime polynomial pairs of arbitrary degree and for all pairs of degree at most two. For A⊂N and m∈Z, let dA(m) count the members of A dividing m. For coprime nonzero F,G∈Z[x], universally unbounded separation between dA(F(n)) and dA(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.
Matija Kazalicki, Siniša Slijepčević
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.
Jonathan G. Bradley-Thrush, Frank Garvan, Jayashree Kalita +1
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 m2−Dn2=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.
Wan Lee, Donghyeok Lim
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 Zp-extensions of number fields. We also extend our study to the case where the group of p-power roots of unity is nontrivial.
Zhiyuan Yang
In 1986, Bombieri, Friedlander and Iwaniec famously obtained that primes are equidistributed in arithmetic progressions to moduli up to x4/7−ε, using well-factorable weights. In this paper, we apply Pascadi's triply-well-factorable convolution estimate and his estimation of incomplete Kloosterman sums to generalize this result to a convolutionform, which improves Wang's result under certain conditions. As for application, we consider the asymptotic density of #{n≤x:P+(n)<P+(n+1)} and #{p≤x:P+(p−1)≥pc}, where P+(n) denote the largest prime factor of n. We show that for x→∞, one hasalign* #{n≤ x:P^+(n)<P^+(n+1)}>0.296x align* and align* &_x→∞ (1/π(x))#{p≤ x:P^+(p-1)≥ p^c}\&≤ S(c)={ aligned & ∫₀¹-c(2/(5/8-189u/200)(1-u))du, && (184/189)≤ c<1,\& S((184/189))+(10/3)(184/189c), && 0.7404<c≤ (184/189), aligned. align* where the function S(c) satisfties S(c)<27logc1 for 0.7404<c<1. The first result improves a previous result 0.280 by the author (2026). The second result constitutes an improvement upon that of Ding and Wang (2025), who obatined limsupx→∞π(x)1#{p≤x:P+(p−1)≥pc}≤27logc1.
Krishnarjun Krishnamoorthy
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.
Yuhang Shi
The spectral side of the Kuznetsov trace formula for GL(2) is governed by the Bessel-Kuznetsov integral transform φˇ(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(αx). In this paper, we provide a rigorous and explicit asymptotic analysis of φˇ(t) in the semiclassical limit t→∞ 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 α. We prove that in the sub-critical regime (α≤1/2π), the transform decays rapidly. Conversely, in the super-critical regime (α>1/2π), the geometric oscillations resonate with the spectral kernel, yielding a localized main term of order O(t−1) with a remarkably simplified arithmetic phase.
Ben Savoie
For each r∈{4,6}, we construct an explicit one-parameter family of elliptic curves over Q(i) containing infinitely many pairwise nonisomorphic curves genuinely defined over Q(i) with j-invariant 1728 and rank exactly r. We construct explicit 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 RM(1,3), which also occurs as the kernel of the quadratic-residue Laplacian governing the Selmer group.
Ian Short, Margaret Stanier, Matty van Son +1
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.
Zhi-Hong Sun
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 gn(x) and vn(x) given by align* &g₀(x)=1,\ g₁(x)=x+12,\ (n+1)²g_n+1(x)=(2n(n+1)+x+12)g_n(x)-n²g_n-1(x)\ (n≥ 1), \&v₀(x)=1,\ v₁(x)=x,\ (n+1)³v_n+1(x)=(2n+1)(n(n+1)+x)v_n(x)-n³v_n-1(x)\ (n≥ 1).align*
Kritika Aggarwal, Debika Banerjee, Shubham Gupta
In this paper, we establish the B4-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 B2-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 B4. 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.
Stefano Vigni
Let f be a non-CM newform of weight k≥4, level N and trivial Nebentypus. Let p∤N be an odd prime number that is ordinary for f and denote by 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 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.
Emanuel Milman
We record several elementary relations between spectral and isoperimetric parameters of a flat torus TΛ=Rn/Λ and the covariance structure of a fundamental domain K for the lattice Λ⊂Rn. For every measurable fundamental domain K and nonzero vector ξ in the dual lattice Λ∗, we observe the sharp directional variance estimate ⟨CovKξ,ξ⟩≥121. This yields a lower bound on the torus spectral gap λSG(TΛ) (equivalently, the length of the shortest nonzero dual vector λ1(Λ∗)) in terms of the maximal covariance of K: λSG(TΛ)=4π2λ1(Λ∗)2≥3∥CovK∥opπ2. Analogous sharp results are obtained for the isoperimetric profile and the Cheeger constant DChe(TΛ) using an old argument of Hadwiger. In particular, when the Voronoi cell KΛ of a lattice with detΛ=1 is isotropic, the recent resolution of the Slicing Problem by Klartag and Lehec implies that DChe(TΛ),λSG(TΛ),λ1(Λ∗)≥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 Λ and corresponding Voronoi cells K=KΛ 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).
Taekyun Kim, Dae san Kim
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
Hai-Liang Wu, He-Xia Ni
Let f(t)∈Z[t] be a nonconstant polynomial with nonzero discriminant and let k≥2 be an integer. For every sufficiently large prime p≡1(modk), by applying mixed exponential sums over finite fields, discrete Fourier analysis and the spectral graph theory, we establish a threshold c(p,k,f) such that any subset A⊆Fp with #A≥c(p,k,f) admits a permutation a1,a2,⋯,a#A of A satisfying f(ai+ai+1)∈{xk:x∈Fp∗} for any 1≤i≤#A, where a#A+1=a1. Also, we give lower and upper bounds for the least possible threshold.
Bernardo San Martín, Víctor Sirvent
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 pk,r(x)=xk−xk−1−⋯−xr+xr−1+⋯+x+1, where k≥2r+1, and k≥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.
Yeongseong Jo, Akash Yadav
We prove that the archimedean Asai L-factor attached to an irreducible generic representation of GLn(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 GLm(F) can be expressed as a finite sum of Jacquet--Shalika local zeta integrals.
Hansol Kim
Enrique González-Jiménez and Álvaro Lozano-Robledo proved that there exists a uniform bound on the levels n for which the n-division field Q(E[n])/Q of an elliptic curve E/Q defined over 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.
Takashi Goda
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 P1(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≥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 P1(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.