Let G be a multiplicatively written finite group of order n. The Gao constant of the group G, denoted E(G), is defined as the smallest positive integer ℓ with the following property: for any given sequence g1⋅…⋅gℓ over G, there exist n distinct integers i1,…,in∈{1,…,ℓ} such that the product of gi1,…,gin, in some order, is the identity element of G. This constant originates from the celebrated additive theorem proved by Erdős, Ginzburg and Ziv in 1961, which amounts to proving E(G)≤2|G|−1 holds in case that G is abelian. It is also well-known that E(G)=2|G|−1 holds for all finite cyclic groups. In 2010, Gao and Li [J. Pure Appl. Algebra] conjectured that E(G)≤3|G|2 for every finite non-cyclic group G. In this paper, we confirm the conjecture for all non-cyclic groups G whose order is not divisible by four, and we also characterize the groups achieving the equality E(G)=3|G|2 as those with a cyclic subgroup of index two.
In this paper, we establish average bounds forSX(f,gj;α,β)=∑nλf(n)λgj(n)e(αnβ)ϕ(nX) over the spectral family, where f is a fixed Hecke cusp form (Maass or holomorphic) for SL2(Z), and {gj} is an orthonormal basis of Hecke–Maass cusp forms with spectral parameters tj, 0<β≤1 and α≠0 are fixed real numbers, ϕ is a smooth function with compact support (1,2). We derive estimates that improve upon the resonance barrier on average. This work provides insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec, Luo, and Sarnak.
We study sets N of the prime numbers defined by combinations of arithmetic constraints. For a fixed real number alpha, we investigate the uniform distribution modulo one of the sequence (alpha p)(p is an element of N). (c) 2026 Published by Elsevier Inc.
We consider the simultaneous Pell equationsx2−ay2=1,z2−bx2=1, where a≥b≥2 are positive integers. We describe a procedure which, for any fixed b, either confirms that the simultaneous Pell equations have at most one solution in positive integers, or finds all exceptions for which we have two solutions.
Bringmann, Guerzhoy and Kane introduced the notion of traces of cycle integrals of harmonic weak Maass form of negative weight in terms of traces of weak Maass form of weight 0. They proved that these traces coincide with the traces of cycle integrals of the associated classical cusp forms. Their result applies to the case of positive discriminants d,δ such that dδ not a square. In this paper, we define traces of cycle integral of weak Maass forms of weight 0 in the case of negative discriminants d,δ such that dδ not a square and prove the similar result. Further, we define modified traces of weak Maass forms of weight 0 in the case of discriminants d,δ such that dδ a square and prove that for a given harmonic weak Maass form of negative weight, the modified traces of cycle integral of the associated weak Maass form of weight 0 coincide with the twisted central L-values of the associated classical cusp forms. As an application of our result, we classify all weakly holomorphic modular forms lying in the space of harmonic weak cusp forms in terms of vanishing of modified traces (twisted central L-value).
We study the two-color distinct-part series S_1(q), equivalently Andrews' generating function v_d(q) for strictly concave compositions, and its odd and even companions T_o(q) and T_e(q). We determine the coefficients of S_1(q) modulo 4 and obtain a complete criterion for the resulting Ramanujan-type progressions. For the even companion, we give a direct overpartition interpretation of its coefficients and show that two natural partition families are each counted by half of those coefficients. For the eta-normalized odd companion C(q)=(q;q)_∞ T_o(q), we prove a quintic self-similarity, derive exact vanishing relations and infinite sign changes for its coefficients, and show that c(n) can be nonzero only when 24n+28 is represented by x^2+3y^2.
We study how Rankin-Selberg periods and distinction problems interact with integral structures in spherical Whittaker type representations. Using this representation-theoretic framework, we settle a conjecture of Loeffler by showing that the local Euler factors appearing in the construction of the motivic Rankin-Selberg Euler system for a product of modular forms are integrally optimal; i.e. any construction of this type with any choice of integral input data in the recipe of Loeffler-Skinner-Zerbes, would give local factors appearing in tame norm relations at p, which are integrally divisible by the Euler factor P-p'(Frob(p)(-1)) modulo p-1. We also interpret this as an integrality result on the unramified part of the period associated to the Rankin-Selberg convolution of two modular forms. Crown Copyright (c) 2026 Published by Elsevier Inc.
Let {a(n)} be an infinite sequence of positive integers with a(k) | a(k +1) such that lim sup(k ->infinity) log a(k)/k < +infinity and lim sup(k ->infinity) Sigma(d|ak) 1/d < +infinity. Then the set {p(1)(2) + p(2)(2) + a(k) + a(l) : p(1), p(2) are primes and k, l >= 1} has a positive lower density. (c) 2026 Published by Elsevier Inc.
It is well known that an Eisenstein series can be written as a product of Eisenstein series only in very few particular cases, for example, E-8 = E-4(2), E-10 = E4E6 and E-14 = E4E10. Moreover, it is also known that a Hecke eigenform cannot be written as a product of any other Hecke eigenforms, unless they are forced by the dimension restrictions. Also, it turns out that the equality between two monomials of the Eisenstein series can be reduced to the above-mentioned relations. In this article, we investigate the question of possible equality between two monomials of the Poincar & eacute; cusp forms of index 1. R.A. Rankin proved that for any even integer k >= 4, the zeros of the Poincar & eacute; cusp form of weight k and index 1, in the standard fundamental domain, lie on the arc A := {e(iota theta) : theta is an element of [pi/2, 2 pi/3]}. We establish a separation property of the zeros of these Poincar & eacute; cusp forms on the arc A(degrees) := {e(iota theta) : theta is an element of (pi/2, 2 pi/3)} and prove that two distinct monomials composed of the index 1 Poincar & eacute; cusp forms are never equal when we take the weight of each of these forms >= 36. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We study the family of algebraic curves of genus >= 1 defined by the affine equations y(s) = ax(r )+ b over a number field k, where r >= 2 and s >= 2 are fixed integers. Assuming the strong version of Lang's conjecture on varieties of general type, we prove that the Mordell-Weil rank of the Jacobian varieties of these curves is uniformly bounded. The proof proceeds by constructing a parameter space for curves in the family with a given number of rational points and analyzing the geometry of its fibers, which are shown to be complete intersection curves of increasing genus. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
We connect generalizations of the classical Hurwitz class numbers coming from two different frameworks: one introduced by Pei and Wang, arising from the generalized Cohen–Eisenstein series, and another by Li, Skoruppa, and Zhou, arising from Eichler orders of quaternion algebras. As applications, we obtain new basis for Eisenstein space E3/2+(4N,id), a generalization of recent results of Beckwith and Mono, and a generalization of Gauss' formula.
We extend bounds on additive energies of modular square roots by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu and apply these results to obtain bounds on certain bilinear exponential sums with modular square roots. From here, we make partial progress on the large sieve for square moduli.
In 1982, Schlickewei and Van der Poorten claimed that any multi-recurrence sequence has, essentially, maximal possible growth rate. Fourty years later, Fuchs and Heintze provided a non-effective proof of this statement. In this paper, we prove a quantitative version of that result by giving an explicit upper bound for the maximal possible growth rate of a multi-recurrence. Moreover, we also give a function field analogue of the result, answering a question posed by Fuchs and Heintze when proving a bound on the growth of multi-recurrences in number fields.
It is known that Hilbert's Tenth Problem over the Gaussian ring & Zopf;[i] = {a + bi : a, b is an element of & Zopf;} is undecidable. In this paper we obtain the following further result: There is no algorithm to decide whether an arbitrarily given polynomial equation P(z1, ... , z20) = 0 (with integer coefficients and 20 unknowns) is solvable over & Zopf;[i]. This improves the previous record involving 52 unknowns. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this paper, we study the probability that some weighted partial sums of a random multiplicative function f are positive. Applying the characteristic decomposition, we obtain that if S is a non-empty subset of the multiplicative residue class group (Z/mZ)× with m being a fixed positive integer and A={a+mn|n=0,1,2,3,⋯} with a∈S, then there exists a positive number δ independent of x, such thatP(∑A∩[1,x)f(n)n<0)>δ unless S is a subgroup of (Z/mZ)× such that the elements of (Z/mZ)×/S have order 1 or 2, in which case we haveP(∑A∩[1,x)f(n)n<0)=O(exp(−exp(lnxCloglogx))) for a positive constant C. This includes as a special case a result of Angelo and Xu. We also extend the result to the cyclotomic field Kn=Q(ζn) with ζn=e2πi/n and study the probability that these generalized weighted sums are positive. In addition, we deal with the positivity problem of certain partial sums related to the celebrated Ramanujan tau function τ(n) and the Ramanujan modular form Δ(q), and obtain an upper bound for the probability that these partial sums are negative in a more general situation.
Starting with a primitive Dirichlet character of conductor N, we construct a paramodular Siegel Eisenstein series of level N^2 and weight k≥4. We calculate the Fourier expansion of the holomorphic Siegel modular form thus constructed. The function is a paramodular newform, and its adelization generates an irreducible automorphic representation.
Benford's law is the statement that in many real-world data set, the probability of having digit d in base B as the first digit is logB((d+1)/d) for all 1≤d≤B. We sometimes refer to this as weak Benford behavior, and we say that a data set satisfies strong Benford behavior in base B if the probability of having significand at most s is logB(s) for all s∈[1,B). We examine Benford behaviors in two different probabilistic models: stick and box fragmentation models. Building on the joint work of Becker et al. [2] on the single proportion stick fragmentation model, we employ combinatorial identities on multinomial coefficients to reduce the multi-proportion stick fragmentation model to the single proportion model. We then provide a necessary and sufficient condition for the lengths of the stick fragments to converge to strong Benford behavior along with a quantification of the discrepancy from uniform distribution on [0,1] in terms of irrationality exponent. Then we answer a conjecture posed by Betti et al. [6] on the high dimensional box fragmentation model. Using tools from Fourier analysis and order statistics, we prove that under some conditions, faces of any arbitrary dimension of the box have total volume converging to strong Benford's behavior.
We give an asymptotic formula for the number of rational points of bounded height on algebraic varieties defined by systems of multihomogeneous diagonal equations. The proof uses the Hardy-Littlewood circle method and the hyperbola method developed by Blomer and Brüdern.
In recent work, the author, in collaboration with Allen, Long, and Tu, developed the Explicit Hypergeometric Modularity Method (EHMM), which establishes the modularity of a large class of hypergeometric Galois representations in dimensions two and three. One important application of the EHMM is the construction of an explicit family of eta-quotients, which we call the K2 functions, from the hypergeometric background. In this article, we introduce an analogous family of eta-quotients, which we call the K3 functions. These K3 functions are constructed using the theory of weight one cubic theta functions originally developed by Jonathan and Peter Borwein. We then use the K3 functions in the EHMM to resolve several hypergeometric modularity conjectures of Dawsey and McCarthy. Further, we provide applications to special L-values of the K3 functions and to the study of generalized Paley graphs.
