We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime q and a finite abelian q-group H, we consider the set of integers n≤x such that the Sylow q-subgroup of the multiplicative group (ℤ/nℤ)× is isomorphic to H. We show that the counting function of this

We study some classical identities for multiple zeta values and show that they still hold for zeta functions built from an arbitrary sequence of nonzero complex numbers. We introduce the complementary zeta function of a system, which naturally occurs when lifting identities for multiple zeta values to identities for quasisymmetric functions.

Let τ(n) be Ramanujan’s tau function, defined by the discriminant modular form Δ(z)=q∏j=1∞(1−qj)24=∑n=1∞τ(n)qn,q=e2πiz (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer’s conjecture asserts that τ(n)≠0 for all n≥1; since τ(n) is multiplicative, it suffices to study primes p for which τ(p) might possibly be zero. Assuming standard conjectures for the twisted

Let N/K be a Galois extension of p-adic number fields and let V be a de Rham representation of the absolute Galois group GK of K. In the case V=ℚp(1), the equivariant local 𝜖-constant conjecture describes the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin L-functions and it can be formulated as the vanishing of a certain element RN/K in K0(ℤp[G],ℚpc[G]);

Let f:ℤ+→ℤ+ be a polynomial with the property that corresponding to every prime p there exists an integer ℓ such that p∤f(ℓ). In this paper, we establish some equidistributed results between the number of partitions of an integer n whose parts are taken from the sequence {f(ℓ)}ℓ=1∞ and the number of parts of those partitions which are in a certain arithmetic progression.

We consider the action of Hecke-type operators on Hilbert–Siegel theta series attached to lattices of even rank. We show that the average Hilbert–Siegel theta series are eigenforms for these operators, and we explicitly compute the eigenvalues.

Let a,b be fixed positive integers such that a is not a perfect square and b is squarefree, and let ω(b) denote the number of distinct prime divisors of b. Let (u1, v1) denote the least solution of Pell equation u2−av2=1. Further, for any positive integer n, let un=αn+ᾱn2 and vn=αn−ᾱn2a, where α=u1+v1a and ᾱ=u1−v1a. In this paper, using the basic properties of Pell equations and some known results

Let (Fn)n≥1 be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that loglcm(F1,F2,…,Fn)∼3logαπ2⋅n2,as n→+∞, where lcm is the least common multiple and α:=1+5)/2 is the golden ratio. We prove that for every periodic sequence s=(sn)n≥1 in {−1,+1} there exists an effectively computable rational number Cs>0 such that loglcm(F3+s3,F4+s4,…,Fn+sn)∼3logαπ2⋅Cs⋅n2,as n→+∞. Moreover, we show that

This paper describes cubic function fields L/K with prescribed ramification, where K is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely cubic closure K′/K of L/K is of genus zero, and a description of the twists of L/K up to isomorphism over K. For cubic function fields of genus at most one, we also describe

In this paper, we consider the Diophantine equation in the title, where p,q are distinct odd prime numbers and a,b are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of p,q,a and b. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well

The goal of this paper is to define an analogue of the Weil-pairing for Drinfeld modules using explicit formulas and to deduce its main properties from these formulas. Our result generalizes the formula given for rank 2 Drinfeld modules by van der Heiden and works as a more explicit, elementary proof of the Weil-pairing’s existence given by van der Heiden.

An l-regular overpartition of n is an overpartition of n into parts not divisible by l. Let A¯l(n) be the number of l-regular overpartitions of n. Andrews defined singular overpartitions counted by the partition function C¯k,i(n). It denotes the number of overpartitions of n in which no part is divisible by k and only parts≡±i(modk) may be overlined. He proved that C¯3,1(9n+3) and C¯3,1(9n+6) are divisible

We consider the identity component of the Sato–Tate group of the Jacobian of curves of the form C1:y2=x2g+2+c,C2:y2=x2g+1+cx,C3:y2=x2g+1+c, where g is the genus of the curve and c∈ℚ∗ is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for C1 curves of genus 4 and 5 and prove what the identity component of the Sato–Tate group

We prove an explicit version of Burgess’ bound on character sums for composite moduli.

In this note, we prove a conjecture of Reitzes, Vulakh and Young on the location of the zeros of certain cuspforms of level one.

In this paper, we obtain an upper bound for the number of integral solutions, of given height, of system of two quadratic forms in five variables. Our bound is an improvement over the bound given in [H. Iwaniec and R. Munshi, The circle method and pairs of quadratic forms, J. Théor. Nr. Bordx.22 (2010) 403–419].

For any positive integer k≥1, let p1/k(n) be the number of solutions of the equation n=[a1k]+⋯+[adk] with integers a1≥⋯≥ad≥1, where [t] is the integral part of real number t. Recently, Luca and Ralaivaosaona gave an asymptotic formula for p1/2(n). In this paper, we give an asymptotic development of p1/k(n) for all k≥1. Moreover, we prove that the number of such partitions is even (respectively, odd)

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalizing known results for Catalan numbers.

Congruences and related identities are derived for a set of colored and weighted partition functions whose generating functions generate the graded algebra of integer weight modular forms of level seven. The work determines a general strategy for identifying and proving identities and associated congruences for modular forms on the principal congruence subgroup of level 7. Ramanujan’s partition congruence

Let k be a number field, Ok its ring of integers, Cl(k) its classgroup and h the class number of k. Let Γ be a finite group. Let ℳ be a maximal Ok-order in the semi-simple algebra k[Γ] containing Ok[Γ], and Cl(ℳ) its locally free classgroup. Let A=ℤ∪{1m,m∈ℤ∖{0,1,−1}} and n∈A. We define the set ℛ(𝒟n,ℳ) of Galois module classes realizable by the nth power of the different to be the set of classes c∈Cl(ℳ)

Let f(n) and g(n) be the number of unordered and ordered factorizations of n into integers larger than one. Let F(n) and G(n) have the additional restriction that the factors are coprime. We establish asymptotic bounds for the sums of F(n)β and G(n)β up to x for all real β and the asymptotic bounds for f(n)β and g(n)β for all negative β.

We investigate the order of exponential sums involving the coefficients of general L-functions satisfying a suitable functional equation and give some new estimates, including refining certain results in preceding works [X. Ren and Y. Ye, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for GLm(ℤ), Sci. China Math.58(10) (2015) 2105–2124; Y. Jiang and G. Lü, Oscillations

For each algebraic number α∈ℚ̄, a result of Habegger [P. Habegger, Singular moduli that are algebraic units, Algebra Number Theory9(7) (2015) 1515–1524] shows that there are only finitely many singular moduli j such that j−α is an algebraic unit. His result uses Duke’s Equidistribution Theorem and is thus not effective. In this paper, we give an effective proof of Habegger’s result assuming that α

In this paper we prove that for every integer d≥1, there exists an explicit constant Bd such that the following holds. Let K be a number field of degree d, let q>max{d−1,5} be any rational prime that is totally inert in K and E any elliptic curve defined over K such that E has potentially multiplicative reduction at the prime 𝔮 above q. Then for every rational prime p>Bd, E has an irreducible mod

Let d be an odd positive square-free integer. In this paper, we shall investigate the structure of the 2-class group of the cyclotomic ℤ2-extension of the imaginary biquadratic number field ℚ(d,−1) if d is of specifiic form. Furthermore, we deduce the structure of the 2-class group of the cyclotomic ℤ2-extension of ℚ(−d).

Given a weight vector τ=(τ1,…,τn)∈ℝ+n with each τi bounded by certain constraints, we obtain a lower bound for the Hausdorff dimension of the set 𝒲n(τ)∩ℳ, where ℳ is a twice continuously differentiable manifold. From this we produce a lower bound for 𝒲n(Ψ)∩ℳ where Ψ is a general approximation function with certain limits. The proof is based on a technique developed by Beresnevich et al. in 2017,

Recently, Hirschhorn and the first author considered the parity of the function a(n) which counts the number of integer partitions of n wherein each part appears with odd multiplicity. They derived an effective characterization of the parity of a(2m) based solely on properties of m. In this paper, we quickly reprove their result, and then extend it to an explicit characterization of the parity of a(n)

We give two variations on a result of Wilkie’s [A. J. Wilkie, Complex continuations of ℝan,exp-definable unary functions with a diophantine application, J. Lond. Math. Soc. (2) 93(3) (2016) 547–566] on unary functions definable in ℝan,exp that take integer values at positive integers. Provided that the function grows slower (in a suitable sense) than the function 2x, Wilkie showed that it must be eventually

For an algebraic number field K, let ζK(s) be the associated Dedekind zeta-function. It is conjectured that ζK(m) is transcendental for any positive integer m>1. The only known case of this conjecture was proved independently by Siegel and Klingen, namely that, when K is a totally real number field, ζK(2n) is an algebraic multiple of π2n[K:ℚ] and hence, is transcendental. If K is not totally real,

The minimal excludant of a partition λ, mex(λ), is the smallest positive integer that is not a part of λ. For a positive integer n, σmex(n) denotes the sum of the minimal excludants of all partitions of n. Recently, Andrews and Newman obtained a new combinatorial interpretation for σmex(n). They showed, using generating functions, that σmex(n) equals the number of partitions of n into distinct parts

In this paper, we show that the lower density of integers representable as the sum of a prime and a Fibonacci number is at least 0.0254905.

We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel’s mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.

Given a field K and n>1, we say that a polynomial f∈K[x] has newly reducible nth iterate over K if fn−1 is irreducible over K, but fn is not (here fi denotes the ith iterate of f). We pose the problem of characterizing, for given d,n>1, fields K such that there exists f∈K[x] of degree d with newly reducible nth iterate, and the similar problem for fields admitting infinitely many such f. We give results

Let p be an odd prime and q a power of p. We examine the deformation theory of reducible and indecomposable Galois representations ρ̄:Gℚ→GSp2n(𝔽q) that are unramified outside a finite set of primes S and whose image lies in a Borel subgroup. We show that under some additional hypotheses, such representations have geometric lifts to the Witt vectors W(𝔽q). The main theorem of this paper is a higher-dimensional

We show that if {Un}n≥0 is a Lucas sequence, then the largest n suc that |Un|=Cm1Cm2⋯Cmk with 1≤m1≤m2≤⋯≤mk, where Cm is the mth Catalan number satisfies n<6500. In case the roots of the Lucas sequence are real, we have n∈{1,2,3,4,6,8,12}. As a consequence, we show that if {Xn}n≥1 is the sequence of the X coordinates of a Pell equation X2−dY2=±1 with a nonsquare integer d>1, then Xn=Cm implies n=1.

In this paper, we describe a new polynomial factorization algorithm over finite fields with odd characteristics. The main ingredient of the algorithm is special singular curves. The algorithm relies on the extension of the Mumford representation and Cantor’s algorithm to these special singular curves.

In this paper, we establish a very flexible and explicit Voronoï summation formula. This is then used to prove an almost Weyl strength subconvexity result for automorphic L-functions of degree two in the depth aspect. That is, looking at twists by characters of prime power conductor. This is the natural p-adic analogue to the well-studied t-aspect.

We explore the possibilities for the Galois representation ρg attached to a weight-one newform g to be residually reducible, i.e. for the Hecke eigenvalues to be congruent to those of a weight-one Eisenstein series. A special role is played by Eisenstein series E11,ηK of level dK, where ηK is the quadratic character associated with an imaginary quadratic field K, of discriminant dK, with respect to

Recently, the first author together with Jens Marklof studied generalizations of the classical three distance theorem to higher-dimensional toral rotations, giving upper bounds in all dimensions for the corresponding numbers of distances with respect to any flat Riemannian metric. In dimension two they proved a five distance theorem, which is best possible. In this paper, we establish analogous bounds

We examine oscillations in a number of sums of arithmetic functions involving Ω(n), the total number of prime factors of n, and ω(n), the number of distinct prime factors of n. In particular, we examine oscillations in Sα(x)=∑n≤x(−1)n−Ω(n)/nα and in Hα(x)=∑n≤x(−1)ω(n)/nα for α∈[0,1], and in W(x)=∑n≤x(−2)Ω(n). We show for example that each of the inequalities S0(x)<0, S0(x)>3.3x, S1(x)>0, and S1(x)x<−3

We survey divisibility properties of the Fourier coefficients of modular functions inspired by Ramanujan. Then using recent results of the generalized Hecke operator on harmonic Maass functions and known divisibility of Fourier coefficients of modular functions, we establish congruence relations of the Fourier coefficients of certain modular functions and mock modular functions of various levels.

Motivated by certain q-series of Ramanujan, we examine two overpartition difference functions. We give both combinatorial and asymptotic formulas for the differences and show that they are always positive. We also briefly discuss similar differences for some other types of partitions. Our main tools are elementary q-series transformations and Ingham’s Tauberian theorem.

Shimura defined a family of maps from the space of modular forms of half-integral weight to the space of modular forms of integral weight. Selberg in his unpublished work found explicitly this correspondence (the first Shimura map 𝒮1) for the class of forms which are products of a Hecke eigenform of level one and a Jacobi theta function. Later, Cipra generalized the work of Selberg to the case where

We prove that the function Pα(x)=exp(1−x−α) with α>0 is 1/α-mild. We apply this result to obtain a uniform 1/α-mild parametrization of the family of curves {xy=𝜖2|(x,y)∈(0,1)2} for 𝜖∈(0,1), which does not have a uniform 0-mild parametrization by work of Yomdin. More generally, we can parametrize families of power-subanalytic curves. This improves a result of Binyamini and Novikov that gives a 2-mild

Let q be a power of a prime number p, 𝔽q be a finite field with q elements and 𝒢 be a subgroup of (𝔽q,+) of order p. We give an existence criterion and an algorithm for computing maximally 𝒢-fixed c-Wieferich primes in 𝔽q[T]. Using the criterion, we study how c-Wieferich primes behave in 𝔽q[T] extensions.

We give a complete description of all solutions to the equation f13+f23=f33+f43 for quadratic forms fj∈ℂ[x,y] and show how Ramanujan’s example can be extended to three equal sums of pairs of cubes. We also give a complete census in counting the number of ways a sextic p∈ℂ[x,y] can be written as a sum of two cubes. The extreme example is p(x,y)=xy(x4−y4), which has six such representations.

An odd Durfee symbol of n is an array of positive odd integers and a subscript D, a1a2⋯asb1b2⋯btD such that 2D+1≥a1≥a2≥⋯≥as>0, 2D+1≥b1≥b2≥⋯≥bt>0, and n=∑i=1sai+∑j=1tbj+2D2+2D+1. Andrews defined the odd rank of an odd Durfee symbol as (s−t). Let N0(a,M;n) be the number of odd Durfee symbols of n with odd rank congruent to a modulo M. We decompose the generating function of N0(a,M;n) into modular and

By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish, using the method of Zagier and Zwegers on holomorphic projection, that this is indeed the case for certain (twisted) “small divisors” summatory functions σψsm(n). More

Given a pair of distinct unitary cuspidal automorphic representations for GL(n) over a number field, let S denote the set of finite places at which the automorphic representations are unramified and their associated Hecke eigenvalues differ. In this paper, we demonstrate how conjectures on the automorphy and possible cuspidality of adjoint lifts and Rankin–Selberg products imply lower bounds on the

We obtain certain estimates for averages of shifted convolution sums involving the Fourier coefficients of a normalized Hecke–Maass eigenform and holomorphic cusp form.

In this paper, we study the order of the pole of the triple tensor product L-functions L(s,π1×π2×π3,⊗3) for cuspidal automorphic representations πi of GLni(𝔸F) in the setting where one of the πi is a monomial representation. In the view of Brauer theory, this is a natural setting to consider. The results provided in this paper give crucial examples that can be used as a point of reference for Langlands’

In this work, we use the Rankin–Selberg method to obtain results on the analytic properties of the standard L-function attached to vector-valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted L-function beyond the usual range of absolute convergence. Our results include also the case of metaplectic Siegel

In this paper, we investigate the Galois conjugates of a Pisot number q∈(m,m+1), m≥1. In particular, we conjecture that for q∈(1,2) we have |q′|≥5−12 for all conjugates q′ of q. Further, for m≥3, we conjecture that for all Pisot numbers q∈(m,m+1) we have |q′|≥m+1−m2+2m−32. A similar conjecture if made for m=2. We conjecture that all of these bounds are tight. We provide partial supporting evidence

“Quasi-elliptic” functions can be given a ring structure in two different ways, using either ordinary multiplication, or convolution. The map between the corresponding standard bases is calculated. A related structure has appeared recently in the computation of Feynman integrals. The two approaches are related by a sequence of polynomials closely tied to the Eulerian polynomials.

For each prime p≡1(mod4) consider the Legendre character χ=(⋅p). Let p±(n) be the number of partitions of n into parts λ>0 such that χ(λ)=±1. Petersson proved a beautiful limit formula for the ratio of p+(n) to p−(n) as n→∞ expressed in terms of important invariants of the real quadratic field K=ℚ(p). But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian

We prove that there exists positive even integer k such that ϕ(n)=ϕ(n+k) holds for infinitely many n. We also prove various estimates on number of solutions to ϕ(p−1)=ϕ(q−1) for distinct primes p and q.

Based on the general strategy described by Borel and Serre and the Voronoi algorithm for computing unit groups of orders we present an algorithm for finding presentations of S-unit groups of orders. The algorithm is then used for some investigations concerning the congruence subgroup property.

We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parameterizing matrix. This provides evidence towards the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence, or that Khintchine’s theorem still holds when restricted to the subspace. This result is proved as a special

Using a modular equation of level 3 and degree 23 due to Chan and Liaw, we prove the fastest known (conjectured to be the fastest one) convergent rational Ramanujan-type series for 1/π of level 3.

Let (𝕋f,𝔪f) denote the mod p local Hecke algebra attached to a normalized Hecke eigenform f, which is a commutative algebra over some finite field 𝔽q of characteristic p and with residue field 𝔽q. By a result of Carayol we know that, if the residual Galois representation ρ¯f:Gℚ→GL2(𝔽q) is absolutely irreducible, then one can attach to this algebra a Galois representation ρf:Gℚ→GL2(𝕋f) that is