Let 𝒟0:={x+iy|x,y∈ℝ,x,y>0}, and let (L,R) be a pair of Möbius transformations corresponding to SL2(ℕ0) matrices such that L(𝒟0) and R(𝒟0) are disjoint. Given such a pair (called a left–right pair), we can construct a directed graph ℱ(L,R) with vertices 𝒟0 and edges {(z,L(z))}z∈𝒟0∪{(z,R(z))}z∈𝒟0, which is a collection of infinite binary trees. We answer two questions of Nathanson by classifying

Let r,n>1 be integers and q be any prime power q such that r|qn−1. We say that the extension 𝔽qn/𝔽q possesses the line property for r-primitive elements if, for every α,𝜃∈𝔽qn∗, such that 𝔽qn=𝔽q(𝜃), there exists some x∈𝔽q, such that α(𝜃+x) has multiplicative order (qn−1)/r. Likewise, if, in the above definition, α is restricted to the value 1, we say that 𝔽qn/𝔽q possesses the translate property

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

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

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 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).

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

In this work, we provide a complete description of Mahler coefficients of uniformly differentiable functions modulo p. The main techniques are related to properties of van der Put coefficients of specific classes of functions, such as 1-Lipschitz measure-preserving functions and uniformly differentiable functions modulo p. The relation tying Mahler coefficients and van der Put coefficients is also

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,

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 obtain a new proof of Hurwitz’s formula for the Hurwitz zeta function ζ(s,a) beginning with Hermite’s formula. The aim is to reveal a nice connection between ζ(s,a) and a special case of the Lommel function Sμ,ν(z). This connection is used to rephrase a modular-type transformation involving infinite series of Hurwitz zeta function in terms of those involving Lommel functions.

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.

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 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(mod4) then fp(A)≡−(A2+4)(p−1)/4(mod p)if A2+4p=1,(−A2−4)(p−1)/4(mod p)if A2+4p=−1, where (⋅p) denotes the Legendre symbol. We also determine

In this paper, we establish a version of the large sieve inequality with square moduli for imaginary quadratic extensions of rational function fields of odd characteristics.

We prove a version of the Khinchin–Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This correspondence together with a multiple ergodic theorem are used to study rational approximations in several congruence classes simultaneously. The result in this part

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

For fixed coprime positive integers a, b, c with a+b=c and min{a,b,c}≥4, there is a conjecture that the exponential Diophantine equation (an)x+(bn)y=(cn)z has only the positive integer solution (x,y,z)=(1,1,1) for any positive integer n. This is the analogue of Jésmanowicz conjecture. In this paper, we consider the equation (m2+n2)x+(2mn)y=(m+n)2z, where m,n are coprime positive integers, and prove

In this paper, we associate a primitive substitution with a family of non-integer positional number systems with respect to the same base but with different sets of digits. In this way, we generalize the classical Dumont–Thomas numeration which corresponds to one specific case. Therefore, our concept also covers beta-expansions induced by Parry numbers. But we establish links to variants of beta-expansions

For any given non-square integer D≡0,1(mod4), we prove Euclid’s type inequalities for the sequence {qi} of all primes satisfying the Kronecker symbol (D/qi)=−1, i=1,2,…, and give a new criterion on a ternary quadratic form to be irregular as an application, which simplifies Dickson and Jones’s argument in the classification of regular ternary quadratic forms to some extent.

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

In this paper, we consider the angular changes of Fourier coefficients of half integral weight cusp forms and sign changes of q-exponents of generalized modular functions.

In this paper, we obtain some Hecke-type identities by using two q-series expansion formulae. And, the identities can also be proved directly in terms of Bailey pairs. In particular, we show that certain partial theta functions and the theta functions can be expressed in terms of Hecke-type identities.

We formulate a notion of modular form on the double half-plane for half-integral weights and explain its relationship to the usual notion of modular form. The construction we provide is compatible with certain physical considerations due to the second author.

We say that a subset of ℙn(ℝ) is maximally singular if its contains points with ℚ-linearly independent homogenous coordinates whose uniform exponent of simultaneous rational approximation is equal to 1, the maximal possible value. In this paper, we give a criterion which provides many such sets including Grassmannians. We also recover a result of the author and Roy about a class of quadratic hypersurfaces

Let Cn− be the minus quotient of the ideal class group of the nth cyclotomic field. In this paper, first, we show that each finite abelian group appears as a subgroup of Cn− for some n. Second, we show that, for all pairs of integers n and m with n|m, the kernel of the lifting map Cn−→Cm− is contained in the 4-torsion Cn−[4] of Cn−. Such an evaluation of the exponent is an individuality of cyclotomic

Let p≡1(mod3) be a prime and denote by ζ3 a primitive third root of unity. Recently, Lemmermeyer presented a conjecture about 3-class groups of pure cubic fields L=ℚ(p3) and of their normal closures k=ℚ(p3,ζ3). The main goal of this paper is to reduce Lemmermeyer’s conjecture to a problem of unit theory by showing that the conjecture of Lemmermeyer follows from Conjecture 2.9.

Kaplansky conjectured that if two positive-definite ternary quadratic forms have perfectly identical representations over ℤ, they are equivalent over ℤ or constant multiples of regular forms, or is included in either of two families parameterized by ℝ2. Our results aim to clarify the limitations imposed to such a pair by computational and theoretical approaches. First, the result of an exhaustive search

It was proved that, for k=341, every sufficiently large even integer can be represented in the form of eight cubes of primes and k powers of 2. In this paper, we sharpen the value of k to 204.

In this paper, we show an analogue of Dawsey’s formula on Chebotarev densities for finite Galois extensions of ℚ with respect to the Riemann zeta function ζ(ms) for any integer m≥2. Her formula may be viewed as the limit version of ours as m→∞.

Let λ1(n) denote the least invariant factor in the invariant factor decomposition of the multiplicative group Mn=(ℤ/nℤ)×. We give an asymptotic formula, with order of magnitude x/logx, for the counting function of those integers n for which λ1(n)≠2. We also give an asymptotic formula, for any even q≥4, for the counting function of those integers n for which λ1(n)=q. These results require a version

We confirm three conjectures of Guo and Zeng on Franel numbers and binomial coefficients. Furthermore, we prove more refined divisibility results for a large class of polynomial sequences.

Let E be an elliptic curve defined over ℚ of conductor N, c the Manin constant of E, and m the product of Tamagawa numbers of E at prime divisors of N. Let K be an imaginary quadratic field where all prime divisors of N split in K, PK the Heegner point in E(K), and III(E/K) the Shafarevich–Tate group of E over K. Let 2uK be the number of roots of unity contained in K. Gross and Zagier conjectured that

Inspired by the recent work of Guo and Zudilin, we establish several results on q-supercongruences by using a q-analogue of Watson’s 3F2 summation formula. In particular, we give a q-analogue of the (A.2) supercongruence of Van Hamme for any prime p≡3(mod4).

For an integer k≥2, let {Fn(k)}n≥2−k be the k-generalized Fibonacci sequence which starts with 0,…,0,1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c with at least two representations as a difference between a k-generalized Fibonacci number and a power of 3. This paper continues the previous work of the first author

Reeder and Yu have recently given a new construction of a class of supercuspidal representations called epipelagic representations [M. Reeder and J.-K. Yu, Epipelagic representations and invariant theory, J. Amer. Math. Soc.27(2) (2014) 437–477, MR 3164986]. We explicitly calculate the Local Langlands Correspondence for certain families of epipelagic representations of unitary groups, following the

In 1851, Prouhet showed that when N=jk+1 where j and k are positive integers, j≥2, the first N consecutive positive integers can be separated into j sets, each set containing jk integers, such that the sum of the rth powers of the members of each set is the same for r=1,2,…,k. In this paper, we show that even when N has the much smaller value 2jk, the first N consecutive positive integers can be separated

Good’s Theorem for regular continued fraction states that the set of real numbers [a0;a1,a2,…] such that limn→∞an=∞ has Hausdorff dimension 12. We show an analogous result for the complex plane and Hurwitz Continued Fractions: the set of complex numbers whose Hurwitz Continued fraction [a0;a1,a2,…] satisfies limn→∞|an|=∞ has Hausdorff dimension 1, half of the ambient space’s dimension.

There are two types of Belyi’s Theorems for curves defined over finite fields of characteristic p, namely the Wild and the Tame p-Belyi Theorems. In this paper, we discuss them in the language of function fields. In particular, we provide a constructive proof for the existence of a pseudo-tame element introduced in [Y. Sugiyama and S. Yasuda, Belyi’s theorem in characteristic two, Compos. Math. 156(2)

Let PRIMES be the set of all primes. A function f:ℕ→ℂ is called multiplicative if f(1)=1 and f(ab)=f(a)f(b) when gcd(a,b)=1. We show that a multiplicative function f which satisfies f(p+q−2)=f(p)+f(q)−f(2)for p,q∈PRIMES satisfies one of the following: (1) f is the identity function, (2) f is the constant function with f(n)=1, (3) f(n)=0 for n≥2 unless n is odd and squareful. As a consequence, a multiplicative

We prove that for the Steinhaus Random Variable z(n), 𝔼∑n∈EN,mz(n)6≍|EN,m|3for m≪(loglogN)13, where EN,m:={1≤n≤N:Ω(n)=m} and Ω(n) denotes the number of prime factors of n.

Let F be a non-archimedean local field of characteristic not equal to 2 and let E/F be a quadratic algebra. We prove the stability of local factors attached to irreducible admissible (complex) representations of GL(2,E) via the Rankin–Selberg method under highly ramified twists. This includes both the Asai as well as the Rankin–Selberg local factors attached to pairs. Our method relies on expressing

Let f(n) be an arithmetic function with f(1)≠0 and let f−1(n) be its reciprocal with respect to the Dirichlet convolution. We study the asymptotic behavior of |f−1(n)| with regard to the asymptotic behavior of |f(n)| assuming that the latter one grows or decays with at most polynomial or exponential speed. As a by-product, we obtain simple but constructive upper bounds for the number of ordered factorizations

We examine “partition zeta functions” analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties — those summed over partitions of fixed length — which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family

In this work, we provide a complete description of Mahler coefficients of 1-Lipschitz measure-preserving functions on the ring of p-adic integers ℤp. Our techniques are mainly based on some congruence identities including binomial coefficients. The main result provides an answer to one of Jeong’s conjectures, concerning a characterization of 1-Lipschitz measure-preserving functions by means of their

Let f(t) be an arbitrary real-valued positive nondecreasing function, in this paper we prove that Sf,k(x)−Tf,k(x)=𝒪(f(x)x131/416(logx)26947/8320),Sf,k(x)−Tf,k(x)=Sf,1(x)−Tf,1(x)+𝒪(f(x)x227/796+𝜀), where Sf,k(x) is the sum given in the title, Tf,k(x)=∫1xf(t){xt}kdt and k is a positive integer. This improves the result of Mercier and Nowak.

For positive integers a, b and c, let T(a,b,c;n) denote the number of representations of a nonnegative integer n as ax1(x1+1)/2+bx2(x2+1)/2+cx3(x3+1)/2 where x1, x2 and x3 are nonnegative integers, and let N(a,b,c;n) denote the number of representations of n as ax12+bx22+cx32 where x1, x2 and x3 are integers. Recently, Sun proved a number of relations between T(a,b,c;n) and N(a,b,c;n) along with numerous

We prove the p-adic version of the Coates–Sinnott Conjecture for all primes p, without assuming the vanishing of μ-invariants, for finite abelian extensions E of a totally real number field F, where either the integral group ring ℤp[G] of the Galois group G is a maximal order in ℚp[G] or E is a CM-field of degree 2m with m odd and p=2, where the group ring ℤ2[G] is not a maximal order. The only assumption

Let E be an elliptic curve defined over ℚ, and let G be the torsion group E(K)tors for some cubic field K which does not occur over ℚ. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) G can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves E/ℚ

We use the theta lifts between Mp2 and PD× to study the distinction problems for the pair (Mp2(E),SL2(F)), where E is a quadratic field extension over a nonarchimedean local field F of characteristic zero and D is a quaternion algebra. With a similar strategy, we give a conjectural formula for the multiplicity of distinction problem related to the pair (Mp2n(E),Sp2n(F)).

Let f be a modular form with complex multiplication. If f has critical slope, then Coleman’s classicality theorem implies that there is a p-adic overconvergent generalized Hecke eigenform with the same Hecke eigenvalues as f. We give a formula for the Fourier coefficients of this generalized Hecke eigenform. We also investigate the dimension of the generalized Hecke eigenspace of p-adic overconvergent

In this paper, we present a description of the ℓ-adic Galois representation attached to an elliptic curve defined over a 2-adic field K, in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely Q8 and SL2(𝔽3), and in each case, we need to distinguish whether the inertia degree of K over ℚ2 is even or odd. The results presented here are being

We study the properties of a sequence cn defined by the recursive relation c0n+1+c1n+2+⋯+cn2n+1=0 for n≥1 and c0=1. This sequence also has an alternative definition in terms of certain norm minimization in the space L2([0,1]). We prove estimates on the growth order of cn and the sequence of its partial sums, infinite series identities, connecting cn with the harmonic numbers Hn and also formulate some