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.

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 Selberg in his unpublished work found explicitly this correspondence (the first Shimura map 𝒮1) for the class of forms which are products of a Hecke eigenform of level one and a Jacobi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

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

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

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’

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.

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

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

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

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 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.

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.

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

We introduce combinatorial interpretations of the coefficients of two second-order mock theta functions. Then, we provide a bijection that relates the two combinatorial interpretations for each function. By studying other special cases of the multivariate identity proved by the bijection, we obtain new combinatorial interpretations for the coefficients of Watson’s third-order mock theta function ω(q)

For β>1, a sequence (cn)n≥1∈ℤℕ+ with 0≤cn<β is the beta expansion of x with respect to β if x=∑n=1∞cnβ−n. Defining dβ(x) to be the greedy beta expansion of x with respect to β, it is known that dβ(1) is eventually periodic as long as β is a Pisot number. It is conjectured that the same is true for Salem numbers, but is only currently known to be true for Salem numbers of degree 4. Heuristic arguments

For an algebraic number field K and a prime number p, let K̃/K be the maximal multiple ℤp-extension. Greenberg’s generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-p extension of K̃ is pseudo-null over the completed group ring ℤp[[Gal(K̃/K)]]. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime

Long and Ramakrishna [Some supercongruences occurring in truncated hyper- geometric series, Adv. Math.290 (2016) 773–808] generalized the (H.2) supercongruence of Van Hamme to the modulus p3 case. In this paper, we give a q-analogue of Long and Ramakrishna’s result for p≡3(mod4). A q-congruence modulo the fourth power of a cyclotomic polynomial, which is a deeper q-analogue of the (A.2) supercongruence

The inequality between rank and crank moments was conjectured and later proved by Garvan himself in 2011. Recently, Dixit and the authors introduced finite analogues of rank and crank moments for vector partitions while deriving a finite analogue of Andrews’ famous identity for smallest parts function. In the same paper, they also conjectured an inequality between finite analogues of rank and crank

We prove a new mock theta function identity related to the partition rank modulo 3 and 9. As a consequence, we obtain the 3-dissection of the rank generating function modulo 9. We also evaluate all of the components of the rank–crank differences modulo 9. These are analogous to conjectures of Lewis [The generating functions of the rank and crank modulo 8, Ramanujan J.18 (2009) 121–146] on rank–crank

We prove that the equation (x−y)4+x4+(x+y)4=zn has no integer solutions x,y,z with gcd(x,y)=1 for all integers n>1. We mainly use a modular approach with two Frey ℚ-curves defined over the field ℚ(30).

Let F be a non-Archimedean local field of characteristic zero. Let X be the p-adic symmetric space X=H\G, where G=GL2n(F) and H=GLn(F)×GLn(F). We verify a conjecture of Sakellaridis and Venkatesh on the Langlands parameters of certain representations in the discrete spectrum of X.

Let K be a number field, and let α1,…,αr be elements of K× which generate a subgroup of K× of rank r. Consider the cyclotomic-Kummer extensions of K given by K(ζn,α1n1,…,αrnr), where ni divides n for all i. There is an integer x such that these extensions have maximal degree over K(ζg,α1g1,…,αrgr), where g=gcd(n,x) and gi=gcd(ni,x). We prove that the constant x is computable. This result reduces to

We illustrate the power of Experimental Mathematics and Symbolic Computation to suggest irrationality proofs of natural constants, and the determination of their irrationality measures. Sometimes such proofs can be fully automated, but sometimes there is still need for a human touch.

Let Γ be a finite rank subgroup of ℚ¯∗. We prove that the multiplicative group of the field generated by all elements in the divisible hull of Γ is free abelian modulo this divisible hull. This proves that a necessary condition for Rémond’s generalized Lehmer conjecture is satisfied.

For a positive integer N, we say that ∞ is a Weierstrass point on the modular curve X0(N) if there is a non-zero cusp form of weight 2 on Γ0(N) which vanishes at ∞ to order greater than the genus of X0(N). If p is a prime with p∤N, Ogg proved that ∞ is not a Weierstrass point on X0(pN) if the genus of X0(N) is 0. We prove a similar result for even weights k≥4. We also study the space of weight k cusp

We consider (k,j)-colored partitions, partitions in which k colors exist but at most j colors may be chosen per size of part. In particular these generalize overpartitions. Advancing previous work, we find new congruences, including in previously unexplored cases where k and j are not coprime, as well as some noncongruences. As a useful aside, we give the apparently new generating function for the

In this paper, we revisit an identity which was proven by Ramanujan and from which he deduced the famous identities that are named after him and L. J. Rogers. Unlike Ramanujan’s proof (which uses the method of q-difference equations), we examine directly the q-coefficients involved. We isolate and identify terms that cancel each other. Once these terms are paired up and canceled, we only need the geometric

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 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

We revisit Munshi’s proof of the t-aspect subconvex bound for GL(3)L-functions, and we are able to remove the “conductor lowering” trick. This simplification along with a more careful stationary phase analysis allows us to improve Munshi’s bound to L(1/2+it,π)≪π,𝜖(1+|t|)3/4−3/40+𝜖.

For x∈[0,1), let [d1(x),d2(x),…] be its Lüroth expansion and {pn(x)qn(x),n≥1} be the sequence of convergents of x. Define the sets 𝜀2(φ)={x∈[0,1):dn+1(x)dn(x)≥φ(n) for infinitely many n∈ℕ},U∗(τ)=x∈[0,1):x−pn(x)qn(x)<1qn(x)(τ+1) for n∈ℕ ultimately and F(τ)=x∈[0,1):limn→∞log(dn(x)dn+1(x))logqn(x)=τ, where φ:ℕ→[2,∞) is a positive function. In this paper, we calculate the Lebesgue measure of the set 𝜀2(φ)

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

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

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

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.