Drinfeld modules and A-motives are the function field analogs of elliptic curves and abelian varieties. For both Drinfeld modules and A-motives, one can construct their l-adic Galois representations and ask whether the images are open. For Drinfeld modules, this question has been answered by Richard Pink and his co-authors; however, this question has not been addressed for A-motives. Here, we clarify

Let 𝒮 be a finite cyclic semigroup written additively. An element e of 𝒮 is said to be idempotent if e+e=e. A sequence T over 𝒮 is called idempotent-sum-free provided that no idempotent of 𝒮 can be represented as a sum of one or more terms from T. We prove that an idempotent-sum-free sequence over 𝒮 of length over approximately a half of the size of 𝒮 is well structured. This result generalizes

S⊂{1,2,…,n} is called a Sidon set if a+b are all distinct for any a,b∈S. Let Sn be the largest cardinal number of such S. We are interested in the sum of elements in the Sidon set S. In this paper, we prove that for any 𝜖>0, ∑a∈Sa=12n3/2+O(n111/80+𝜖), where S⊂{1,2,…,n} is a Sidon set and |S|=Sn.

This paper provides a mean value theorem for arithmetic functions f defined by f(n)=∏d|ng(d), where g is an arithmetic function taking values in (0,1] and satisfying some generic conditions. As an application of our main result, we prove that the density μq(n) (respectively, ρq(n)) of normal (respectively, primitive) elements in the finite field extension 𝔽qn of 𝔽q are arithmetic functions of (nonzero)

Let C be a smooth plane curve of degree d≥4 defined over a global field k of characteristic p=0 or p>(d−1)(d−2)/2 (up to an extra condition on Jac(C)). Unless the curve is bielliptic of degree four, we observe that it always admits finitely many quadratic points. We further show that there are only finitely many quadratic extensions k(D) when k is a number field, in which we may have more points of

Let 𝔸[n] be the group of n-torsion points of a commutative algebraic group 𝔸 defined over a number field F. For a prime 𝔭 of F, we let N𝔭(𝔸[n]) be the number of 𝔽𝔭-solutions of the system of polynomial equations defining 𝔸[n] when reduced modulo 𝔭. Here, 𝔽𝔭 is the residue field at 𝔭. Let πF(x) denote the number of primes 𝔭 of F such that N(𝔭)≤x. We then, for algebraic groups of dimension

In this paper, we investigate the sum of distinct parts that appear at least 2 times in all the partitions of n providing new combinatorial interpretations for this sum. A connection with subsets of {1,2,…,n} is given in this context. We provide two different proofs of our results: analytic and combinatorial. In addition, considering the multiplicity of parts in a partition, we provide a combinatorial

One major theme of this paper concerns the expansion of modular forms and functions in terms of fractional (Puiseux) series. This theme is connected with another major theme, holonomic functions and sequences. With particular attention to algorithmic aspects, we study various connections between these two worlds. Applications concern partition congruences, Fricke–Klein relations, irrationality proofs

We describe an extension of Fine’s method of deriving basic hypergeometric series transformations and derive new transformations from the method. Combinatorial proofs of two of the examples are provided.

Hardy and Ramanujan introduced the Circle Method to study the Fourier expansion of certain meromorphic modular forms on the upper complex half-plane. These led to asymptotic results for the partition numbers and proven and unproven formulas for the coefficients of the reciprocals of Eisenstein series Ek, especially of weight 4. Berndt et al. finally proved them all. Recently, Bringmann and Kane generalized

The ℚ¯-algebra of periods was introduced by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of ℚ-rational functions over ℚ-semi-algebraic domains in ℝd. The Kontsevich–Zagier period conjecture affirms that any two different integral expressions of a given period are related by a finite sequence of transformations only using three

Recurrence formulas for generalized poly-Bernoulli polynomials are given. The formula gives a positive answer to a question raised by Kaneko. Further, as applications, annihilation formulas for Arakawa-Kaneko type zeta-functions and a counting formula for lonesum matrices of a certain type are also discussed.

Three transformation formulae are established for the partial sum of Bailey’s well-poised 6ψ6-series. Their particular cases provide q-analogues of Guillera’s two series for π±2 with convergence rate 2764, and for other classical π-related infinite series.

It is conjectured that the regularized double shuffle relations give all algebraic relations among the multiple zeta values, and hence all other algebraic relations should be deduced from the regularized double shuffle relations. In this paper, we provide as many as the relations which can be derived from the regularized double shuffle relations, for example, the weighted sum formula of Guo and Xie

Let E/F be an elliptic curve defined over a number field F. Suppose that E has complex multiplication over F¯, i.e. EndF¯(E)⊗ℚ is an imaginary quadratic field. With the aid of CM theory, we find elliptic curves whose quadratic twists have a constant root number.

We consider the proportion of genus one curves over ℚ of the form z2=f(x,y) where f(x,y)∈ℤ[x,y] is a binary quartic form (or more generally of the form z2+h(x,y)z=f(x,y) where also h(x,y)∈ℤ[x,y] is a binary quadratic form) that have points everywhere locally. We show that the proportion of these curves that are locally soluble, computed as a product of local densities, is approximately 75.96%. We prove

For any x∈[0,1), let Sn(x) be the partial summation of the first n digits in the binary expansion of x and Rn(x) be its run-length function. The classical Borel’s normal number theorem tells us that for almost all x∈[0,1), the limit of Sn(x)/n as n goes to infinity is one half. On the other hand, the Erdös–Rényi limit theorem shows that Rn(x) increases to infinity with the logarithmic speed log2n as

We insert additional variables into Warnaar’s q-analogue of Nicomachus’ identity and other related identities, and compute discriminants with respect to q. Factorization of these discriminants reveals pairs of partitions that conjecturally relate in the manner of Wheatstone. The factorization also yields, conjecturally, families of polynomials with relations to various Molien series, remarkable rational

We establish discrete and continuous log-concavity results for a biparametric extension of the q-numbers and of the q-binomial coefficients. By using classical results for the Jacobi theta function we are able to lift some of our log-concavity results to the elliptic setting. One of our main ingredients is a putatively new lemma involving a multiplicative analogue of Turán’s inequality.

In this paper, we study the Iwasawa λ-invariant of the cyclotomic ℤ2-extension of ℚ(pq), where p,q are distinct odd prime numbers satisfying certain arithmetic conditions. Moreover, we obtain an upper bound of the 2-part of the class number of certain quartic number fields by calculating the Sinnott index explicitly.

Finding the largest size of a partition under certain restrictions has been an interesting subject to study. For example, it is proved by Olsson and Stanton that for two coprime integers s and t, the largest size of an (s,t)-core partition is (s2−1)(t2−1)/24. Xiong found a formula for the largest size of a (t,mt+1)-core partitions with distinct parts. In this paper, we find an explicit formula for

Several new estimates for the 2-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as to new theorems, including a long-standing open conjecture by Lengyel. The estimates and criteria all depend on our previous analysis of powers of 2 in the denominators

Robin’s criterion states that the Riemann hypothesis is equivalent to σ(n)

For each algebraic number α and each positive real number t, the t-metric Mahler measure mt(α) creates an extremal problem whose solution varies depending on the value of t. The second author studied the points t at which the solution changes, called exceptional points forα. Although each algebraic number has only finitely many exceptional points, it is conjectured that, for every N∈ℕ, there exists

In 2012, Chan posed six conjectures on congruences for Appell–Lerch sums and five of them were proved by mathematicians. In this paper, we confirm the remaining conjectural congruence of Chan by utilizing an identity due to Hickerson and Mortenson and the theory of modular forms.

For a positive definite ternary integral quadratic form f, let r(n,f) be the number of representations of an integer n by f. A ternary quadratic form f is said to be a generalized Bell ternary quadratic form if f is isometric to x2+2αy2+2βz2 for some nonnegative integers α,β. In this paper, we give a closed formula for r(n,f) for a generalized Bell ternary quadratic form f(x,y,z)=x2+2αy2+2βz2 with

Let q be a prime of the form 4k+3. Then for any three mutually orthogonal great circles of the sphere x2+y2+z2=q lying on rational planes, at least one of these circles does not contain any rational points.

We prove a kind of bilateral series that terminates to one side (left or right) related to Ramanujan-like series for negative powers of π, and conjecture a type of supercongruences associated with them. We support this conjecture by checking all the cases for many primes. In addition, we are able to prove a few of them from some terminating hypergeometric identities. Finally, we make an intriguing

In this paper, we prove modular identities conjectured by Koike and Somos for the Rogers–Ramanujan functions. Our methods focus on approaches that Ramanujan could have employed, including the theory of modular equations, dissections of identities, and methods derived from the approaches of Watson and Rogers.

We study a family of residual crank generating functions defined on overpartitions, the so-called kth residual cranks. Specifically, the moment generating functions associated to these cranks exhibit quasimodularity properties which are dependent on the choice of k. We also show that the second moments of these cranks admit a combinatoric interpretation as weighted overpartition counts. This interpretation

We study quasimodular forms of depth ≤4 and determine under which conditions they occur as solutions of modular differential equations. Furthermore, we study which modular differential equations have quasimodular solutions. We use these results to investigate extremal quasimodular forms as introduced by M. Kaneko and M. Koike further. Especially, we prove a conjecture stated by these authors concerning

Let K be a global field and let S be a finite set of primes of K containing the Archimedean primes. We generalize the duality theorem for the Néron S-class group of an abelian variety A over K established previously by removing the requirement that the Tate–Shafarevich group of A be finite. We also derive an exact sequence that relates the indicated group associated to the Jacobian variety of a proper

An overpartition of n is a partition of n in which the first occurrence of a number may be overlined. Then, the rank of an overpartition is defined as its largest part minus its number of parts. Let N¯(s,m,n) be the number of overpartitions of n with rank congruent to s modulo m. In this paper, we study the rank differences of overpartitions N¯(s,m,mn+d)−N¯(t,m,mn+d) for m=4 or 8 and 0≤d,s,t

We give a new proof of the norm relations for the Asai–Flach Euler system built by Lei–Loeffler–Zerbes. More precisely, we redefine Asai–Flach classes in the language used by Loeffler–Skinner–Zerbes for Lemma–Eisenstein classes and prove both the vertical and the tame norm relations using local zeta integrals. These Euler system norm relations for the Asai representation attached to a Hilbert modular

There are many results in the literature concerning power values, equal values or more generally, polynomial values of lattice point counting polynomials. In this paper, we prove various finiteness results for polynomial values of polynomials counting the lattice points on the surface of an n-dimensional cube, pyramid and simplex.

The local sum conjecture is a variant of some of Igusa’s questions on exponential sums put forward by Denef and Sperber. In a remarkable paper by Cluckers, Mustata and Nguyen, this conjecture has been established in all dimensions, using sophisticated, powerful techniques from a research area blending algebraic geometry with ideas from logic. The purpose of this paper is to give an elementary proof

We say that n nonzero ideals of algebraic integers in a fixed number ring are k-wise relatively r-prime if any k of them are relatively r-prime. In this paper, we provide an exact formula for the probability that n nonzero ideals of algebraic integers in a fixed number ring are k-wise relatively r-prime.

We present a family of algorithms for computing the Galois group of a polynomial defined over a p-adic field. Apart from the “naive” algorithm, these are the first general algorithms for this task. As an application, we compute the Galois groups of all totally ramified extensions of ℚ2 of degrees 18, 20 and 22, tables of which are available online.

We evaluate some finite and infinite sums involving q-trigonometric and q-digamma functions. Upon letting q approach 1, one obtains corresponding sums for the classical trigonometric and the digamma functions. Our key argument is a theta product formula of Jacobi and Gosper’s q-trigonometric identities.

In this paper, we study some products related to quadratic residues and quartic residues modulo primes. Let p be an odd prime and let A be any integer. We determine completely the product fp(A):=∏1≤i,j≤(p−1)/2p∤i2−Aij−j2(i2−Aij−j2) modulo p; for example, if p≡1(mod 4) then fp(A)≡−(A2+4)(p−1)/4(mod p)ifA2+4p=1,(−A2−4)(p−1)/4(mod p)ifA2+4p=−1, where (⋅p) denotes the Legendre symbol. We also determine

Polylogarithms are those multiple polylogarithms that factor through a certain quotient of the de Rham fundamental group of the thrice punctured line known as the polylogarithmic quotient. Building on work of Dan-Cohen, Wewers, and Brown, we push the computational boundary of our explicit motivic version of Kim’s method in the case of the thrice punctured line over an open subscheme of Spec ℤ. To do

For primitive nontrivial Dirichlet characters χ1 and χ2, we study the weight zero newform Eisenstein series Eχ1,χ2(z,s) at s=1. The holomorphic part of this function has a transformation rule that we express in finite terms as a generalized Dedekind sum. This gives rise to the explicit construction (in finite terms) of elements of H1(Γ0(N),ℂ). We also give a short proof of the reciprocity formula for

Continued fraction expansions of automatic numbers have been extensively studied during the last few decades. The research interests are, on one hand, in the degree or automaticity of the partial quotients following the seminal paper of Baum and Sweet in 1976, and on the other hand, in calculating the Hankel determinants and irrationality exponents, as one can find in the works of Allouche–Peyrière–Wen–Wen

Let G be a finitely generated multiplicative subgroup of ℚ× having rank r. The ratio between nr and the Kummer degree [ℚ(ζm,Gn):ℚ(ζm)], where n divides m, is bounded independently of n and m. We prove that there exist integers m0,n0 such that the above ratio depends only on G, gcd(m,m0), and gcd(n,n0). Our results are very explicit and they yield an algorithm that provides formulas for all the above

We study a continued fraction due to Ramanujan, that he recorded as Entry 12 in Chapter 16 of his second notebook. It is presented in Part III of Berndt’s volumes on Ramanujan’s notebooks. We give two alternate approaches to proving Ramanujan’s Entry 12, one using a method of Euler, and another using the theory of orthogonal polynomials. We consider a natural generalization of Entry 12 suggested by

In this paper, we show that the iterated integrals on products of one variable multiple polylogarithms from 0 to 1 are actually in the algebra of multiple zeta values if they are convergent. In the divergent case, we define the regularized iterated integrals from 0 to 1. By the same method, we show that the regularized iterated integrals are also in the algebra of multiple zeta values. As an application

The Markoff spectrum is defined as the set of normalized values of arithmetic minima of indefinite quadratic forms. In the theory of the Markoff spectrum we observe various kinds of symmetry. Each of Conway’s topographs of quadratic forms which give values in the discrete part of the Markoff spectrum has a special infinite path consisting of edges. It has symmetry with respect to a translation along

In 2009, Bringmann, Lovejoy and Osburn defined two analogues of the crank function for overpartitions, namely the first residual crank and second residual crank. For a positive integer d, we introduce the notion of a dth residual crank function for overpartitions, examine the positive moments of these generalized crank functions while varying d, and prove some inequalities.

We extend two results by Boxall and Jones on algebraic values of certain analytic functions to meromorphic functions. We obtain C(logH)η bounds for the number of algebraic points of height at most H on certain restrictions of the graphs of such functions. The constant C and exponent η depend on certain data associated with the functions and can be effectively computed from them.

We show that all triples (x1,x2,x3) of singular moduli satisfying x1x2x3∈ℚ× are “trivial”. That is, either x1,x2,x3∈ℚ; some xi∈ℚ and the remaining xj,xk are distinct, of degree 2, and conjugate over ℚ; or x1,x2,x3 are pairwise distinct, of degree 3, and conjugate over ℚ. This theorem is best possible and is the natural three-dimensional analogue of a result of Bilu, Luca and Pizarro-Madariaga in two

We set up a combinatorial framework for inclusion-exclusion on the partitions into distinct parts to obtain an alternative generating function of partitions into distinct and non-consecutive parts. In connection with Rogers–Ramanujan identities, the generating function yields two formulas in Slater’s list. The same formulas were constructed by Hirschhorn. Similar formulas were obtained by Bringmann

For every positive integer k, it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in k arithmetic progressions. For k=1, all forms with this property are determined.

In 1920, Ramanujan studied the asymptotic differences between his mock theta functions and modular theta functions, as q tends towards roots of unity singularities radially from within the unit disk. In 2013, the bounded asymptotic differences predicted by Ramanujan with respect to his mock theta function f(q) were established by Ono, Rhoades, and the author, as a special case of a more general result

We give a combinatorial proof of a generalization of an identity involving the sum of divisors function σ(n) and the partition function p(n), which is a companion of Euler’s recurrence formula for σ(n).

In this paper, we express three different, yet related, character sums in terms of generalized Bernoulli numbers. Two of these sums are generalizations of sums introduced and studied by Berndt and Arakawa–Ibukiyama–Kaneko in the context of the theory of modular forms. A third sum generalizes a sum already studied by Ramanujan in the context of theta function identities. Our methods are elementary,

Dyson’s famous conjectures (proved by Atkin and Swinnerton-Dyer) gave a combinatorial interpretation of Ramanujan’s congruences for the partition function. The proofs of these results center on one of the universal mock theta functions that generate partitions according to Dyson’s rank. George Beck has generalized the study of partition function congruences related to rank by considering the total

Given an integer base b≥2, a number ρ≥1 of colors, and a finite sequence Λ=(λ1,…,λρ) of positive integers, we introduce the concept of a Λ-restricted ρ-colored b-ary partition of an integer n≥1. We also define a sequence of polynomials in λ1+⋯+λρ variables, and prove that the nth polynomial characterizes all Λ-restricted ρ-colored b-ary partitions of n. In the process, we define a recurrence relation

We establish the triple integral evaluation ∫1∞∫01∫01dzdydxx(x+y)(x+y+z)=524ζ(3), as well as the equivalent polylogarithmic double sum ∑k=1∞∑j=k∞(−1)k−1k21j2j=1324ζ(3). This double sum is related to, but less approachable than, similar sums studied by Ramanujan. It is also reminiscent of Euler’s formula ζ(2,1)=ζ(3), which is the simplest instance of duality of multiple polylogarithms. We review this

We formulate and prove an analogue of the non-commutative Iwasawa Main Conjecture for ℓ-adic representations of the Galois group of a function field of characteristic p. We also prove a functional equation for the resulting non-commutative L-functions. As corollaries, we obtain non-commutative generalizations of the main conjecture for Picard-1-motives of Greither and Popescu and a main conjecture

We consider equations of the form Fn=P(x), where P is a polynomial with integral coefficients and Fn is the nth Fibonacci number that is, F0=0,F1=1 and Fn=Fn−1+Fn−2 for n>1. In particular, for each k∈ℕ+, we prove the existence of a polynomial Fk∈ℤ[x] of degree 2k−1 such that the Diophantine equation Fk(x)=Fm has infinitely many solutions in positive integers (x,m). Moreover, we present results of our