In this paper, by making use of the creative microscoping method introduced by Guo and Zudilin [Adv. Math. 346 (2019), 329-358] and the Chinese remainder theorem for coprime polynomials, we establish partial q-analogues of Swisher’s conjectures (A.3) and (D.3), thereby partially confirming them. In particular, we respectively obtain weakened forms of the conjectures proposed by Guo and Zudilin, and by Liu and Wang.
In this paper we study the relationship between isogeny classes of toroidal groups and non-totally real number fields. Under some assumptions concerning the discriminant, we prove that any number field with one pair of complex embeddings arises as the ring of endomorphisms (tensorized with [Formula: see text]) of a toroidal group of complex and real ranks differing by one. On the other hand, any toroidal group [Formula: see text], with [Formula: see text], such that the field [Formula: see text] is of the aforementioned type, arises from [Formula: see text]. In particular, a representation of the toroidal group [Formula: see text] arising from the field [Formula: see text] is provided through a period matrix in standard coordinates of [Formula: see text], expressed in terms of the roots and the coefficients of (a multiple of) the minimal polynomial of a suitable primitive element of [Formula: see text]. Furthermore, for such a toroidal group we present the analytic and rational representations of its ring of endomorphisms. The same problem is also studied in the more intricate case of a field having many pairs of complex embeddings. We consider in detail the quintic case.
We prove that, for any odd prime [Formula: see text] and [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] for [Formula: see text]. When [Formula: see text], this supercongruence reduces to a result of Guillera and Zudilin in 2012. We shall prove this result by building a [Formula: see text]-analogue of it. The main ingredients of the proof include the method of “creative microscoping” devised by Guo and Zudilin, a quadratic transformation of Rahman, a quadratic transformation of Gasper and Rahman, and Jackson’s [Formula: see text] summation. We also give a similar generalization of a supercongruence of Guo and Zudilin along with its [Formula: see text]-analogue.
Let [Formula: see text] be a Lucas sequence, [Formula: see text] be a prime, and [Formula: see text] be the rank of appearance of [Formula: see text] in [Formula: see text], that is, the least positive integer [Formula: see text] such that [Formula: see text], if it exists. We derive closed-form formulas for the Dirichlet density of primes [Formula: see text] for which [Formula: see text], where [Formula: see text] is a fixed integer. Our results complete the work of Sanna ([Formula: see text]) by covering all [Formula: see text] and all [Formula: see text].
Recently, Andrews and Ghosh Dastidar (Ramanujan J. 69, Art. No. 26, 2026) studied two interesting partition statistics, namely, SOME(n) and DSOME(n), where SOME(n) is the sum of all the odd parts in the partitions of n minus the sum of all the even parts, and DSOME(n) is the sum of all the odd parts in the partitions of n into distinct parts minus the sum of all the even parts. They expressed the generating functions of SOME(n) and DSOME(n) in terms of q-series and found several interesting congruences modulo 4 and 5. In this paper, we explore similar statistics for some colored partitions which also satisfy some beautiful congruences.
In this paper we study the number of integer pairs whose products are k-th powers inside a square. A precise asymptotic formula is given.
Lenstra introduced the notion of the Euclidean ideal class, a generalization of the Euclidean domain that captures cyclic class groups. In this article, we establish the existence of Euclidean ideal classes in abelian quartic fields. As a corollary, we demonstrate that certain biquadratic fields with class number two possess a Euclidean ideal class. Additionally, we investigate the presence of Euclidean ideal classes in specific cubic and quadratic extensions.
In our recent paper, “Algorithms for Determination of t-Module Structures on Some Extension Groups,” published in International Journal of Number Theory, Vol. 21, No. 8 (2025) 1889–1922, we identified a gap in Algorithm 2 that affects Example 4.2. In this corrigendum, we describe the nature of this issue and present a corrected version of the algorithm.
Let g be a dimension function and let Psi be an approximation function. The Generalized Baker-Schmidt problem (1970) concerns the g-dimensional Hausdorff measure (H-g-measure) of the set of Psi-approximable points on nondegenerate manifolds. The problem relates the "size" of the set of Psi-approximable points with the convergence or divergence of a certain series. In the dual approximation setting, the divergence case has been established by Beresnevich-Dickinson-Velani (2006). The convergence case, however, represents a challenging open problem, and progress thus far has been effectuated in limited cases only. For instance, for the parabola, it has been established under some restrictions on the dimension function g and for monotonic approximating functions. We prove some related new results for the parabola; in particular, we show that the monotonicity assumption on a multivariable approximating function cannot be removed. We go on to study Veronese curves in higher dimensions. Using Gelfond's lemma and some general irreducibility considerations for integer polynomials, in dimension three we are able to generalize a recent result of Pezzoni (2020) regarding the convergence theory of H-g-measure.
In this paper, we correct the mathematical error of the paper “Critical points of the Eisenstein series of the Fricke group of level 2”.
In this work, we study a continued fractions theory for the topological completion of the field of Puiseux series. As in the classical case, any element in the completion can be uniquely written as a continued fractions, and this approximation is optimal. In this work, we interpret the preceding results in terms of the action of a suitable arithmetic subgroup of the special linear group on the Berkovich space defined over said completion. The quotient space plays a significant role in such description. We also explore the connections between points of type IV of the Berkovich space in terms of some “non-convergent” or “undefined” continued fractions, in a sense that we make precise in the text.
In this paper, we study the limiting distribution of pairs of integers satisfying certain unitary properties. We prove that, for a sufficiently large random sample of integers, the number of semi-unitary coprime pairs converges in distribution to a standard normal random variable, and we provide the rate of convergence. Moreover, we establish a similar result for the limiting distribution of unitary coprime pairs.
In this work, we show that given a finite [Formula: see text]-group [Formula: see text], a number field [Formula: see text] having a trivial [Formula: see text]-class group [Formula: see text], and a finite set of primes [Formula: see text] of [Formula: see text], there exists a finite extension [Formula: see text] such that the [Formula: see text]-split [Formula: see text]-Hilbert class field tower [Formula: see text] of [Formula: see text] has [Formula: see text] as its Galois group. This extends results by Ozaki and Hajir–Maire–Ramakrishna.
In this paper, we study the polynomials [Formula: see text] [Formula: see text], whose rational roots would yield counterexamples to Fermat’s Last Theorem. We investigate their factorization over [Formula: see text]. In the case [Formula: see text], we ask whether they are irreducible over [Formula: see text], prove the irreducibility for several infinite families, and investigate the location of the roots of these polynomials on the complex plane. For [Formula: see text], the factorization of [Formula: see text] is intimately related to that of the Cauchy–Mirimanoff polynomials [Formula: see text] and the polynomials [Formula: see text] and [Formula: see text] introduced by Nanninga. After removing the trivial factors [Formula: see text], [Formula: see text], and [Formula: see text], the remaining components agree (up to change of variable) with [Formula: see text], [Formula: see text], or [Formula: see text]. We prove several new irreducibility results for these factors.
Let [Formula: see text] be a discrete valuation domain. In this paper, we give an estimation for the number of elements of a set [Formula: see text], on which there acts a family of polynomial mappings in a transitive way. This estimation then applies to orbits in [Formula: see text]. Orbits in [Formula: see text] are examined more closely.
Covering systems of the integers were introduced by Erdős in 1950. Since then, many beautiful questions and conjectures about these objects have been posed. Most famously, Erdős asked whether the minimum modulus of a covering system with distinct moduli can be arbitrarily large. This problem was resolved in 2015 by Hough, who proved that the minimum modulus is bounded. In 2022, Balister et al. developed Hough’s method, giving a simpler and more versatile proof of Hough’s result. Their technique has many applications in a number of variants on Erdős’ minimum modulus problem. In this paper, we show some new bounds for Erdős covering systems in global function fields. In particular, we show that there is no covering system of multiplicity s in any global function field of genus g over 𝔽 q for q ≥ (82.26 + 18.88g)e 0.95g s 2 . Moreover, we obtain that there is no covering system of 𝔽 q [x] with distinct moduli for q > 73.
We study the group of inner twists for Siegel modular forms of genus two. For lifted Siegel modular forms (these are Siegel modular forms obtained from elliptic modular forms by various functorial lifts), we compute the group of inner twists in terms of inner twists of elliptic modular forms. On the other hand, for non-lifted Siegel modular forms, we provide necessary and sufficient conditions for the existence of inner twists in terms of the Lie algebras of the images of Galois representations.
Let [Formula: see text] be a positive integer and [Formula: see text] a set of [Formula: see text] distinct positive integers. For a positive integer [Formula: see text], denote by [Formula: see text] and [Formula: see text] the [Formula: see text] matrices whose [Formula: see text]-entries are the [Formula: see text]th powers of the greatest common divisor and the least common multiple of [Formula: see text] and [Formula: see text], respectively. In this paper, we study the divisibility relations among such matrices. Under certain structural conditions on [Formula: see text] (namely, [Formula: see text] is gcd closed and satisfies a specific local condition concerning the maximal size of a certain associated set being [Formula: see text]), we prove that for any positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], the divisibility relations [Formula: see text], [Formula: see text] and [Formula: see text] hold in the ring [Formula: see text] of [Formula: see text] matrices over the integers. This result extends a theorem of Chen, Hong and Zhao from 2022, and in particular, it proves a special case of a conjecture of Hong proposed in [S. F. Hong, Divisibility among power GCD matrices and power LCM matrices, Bull. Aust. Math. Soc. 113 (2026) 231–243].
In recent years, congruence properties for the coefficients of mock theta functions have received a lot of attention. In 2021, Wang gave a systematic study on the parity of coefficients of classical mock theta functions. Recently, Chen and Garvan proved a number of congruences modulo 4 for five mock theta functions. In this paper, we show several congruences modulo 3 and 4 for the sixth order mock theta function psi-(q) defined by Berndt and Chan.
Let zeta(.) denote the Riemann zeta function; let mu(.) and lambda(.) denote the Mobius and Liouville functions, respectively, while M(.) and L(.), respectively, denote their corresponding summatory functions. We consider the correlations (T)= 1/ zeta(1 + delta(T)) Sigma(n <= T) (1-c) mu(n)M(n - 1)/n(1 + delta(T)) and (T)= 1/ zeta(1 + delta(T)) Sigma(n <= T) (1-c) lambda(n)L(n - 1) /n(1+delta(T )) , where 0c-1) is suitably chosen. Under the Riemann hypothesis and simplicity of the nontrivial zeros rho =1/2+ i gamma of zeta(s) we show that (T)= - 3 /pi(2) (1 - T(c-1)delta(T ))+ Sigma(0(T)= 1/2 (1 /zeta(2)(1/2) - 1+ T(c-1)delta(T ) + Sigma(0infinity where 0 <= T(c-1)delta(T ) < 1. These results combined with numerical observations suggest that there is anticorrelation between mu(n) and M(n -1) as well as between lambda(n) and L(n - 1), where the correlation is computed using a logarithmic average. This would imply effective upper bounds on |1/zeta' (rho)|.
