New infinite q-product expansions with vanishing coefficients

The Ramanujan Journal - Motivated by results of Hirschhorn, Tang, and Baruah and Kaur on vanishing coefficients (in arithmetic progressions) in a new class of infinite product which have appeared...     

On infinite arithmetic progressions in sumsets

Let k be a positive integer.Denote by D_1/k the least integer d such that for every set A of nonnegative integers with the lower density 1/k,the set(k+1)A contains an infinite arithmetic progression with difference at most d,where(k+1)A is the set of all sums of k+1 elements(not necessarily distinct) of A.Chen and Li(2019) conjectured that D_1/k=k²+o(k²).The purpose of this paper is to confirm the above conjecture.We also prove that D_1/k is a ...     

Rankin-Selberg coefficients in large arithmetic progressions

Let (λ_f(n))_n≥1 be the Hecke eigenvalues of either a holomorphic Hecke eigencuspform or a Hecke-Maass cusp form f.We prove that,for any fixed η> 0,under the Ramanujan-Petersson conjecture for GL₂ Maass forms,the Rankin-Selberg coefficients(λ_f(n)²)_n≥1 admit a level of distribution θ=2/5+1/260-η in arithmetic progressions.     

Fourier coefficients of GL(N) automorphic forms in arithmetic progressions

We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all \({N \geq 2}\) , satisfy a central limit theorem in a suitable range, generalizing the case N = 2 treated by Fouvry et al. (Commentarii Math Helvetici, 2014). Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.     

The average value of Fourier coefficients of cusp forms in arithmetic progressions

Recently Blomer showed that if α(n) denote the normalized Fourier coefficients of any holomorphic cusp form f with integral weight, then

     

Asymptotics for the Taylor coefficients of certain infinite products

The Ramanujan Journal - Let $$(m_1,\ldots ,m_J)$$ and $$(r_1,\ldots ,r_J)$$ be two sequences of J positive integers satisfying $$1\le r_j< m_j$$ for all $$j=1,\ldots ,J$$ . Let $$(\delta...     

A note on Mertens' formula for arithmetic progressions

We study the Mertens product over primes in arithmetic progressions, and find a uniform version of previous results.     

Matching coefficients in the series expansions of certain q-products and their reciprocals

The Ramanujan Journal - We show that the series expansions of certain q-products have matching coefficients with their reciprocals. Several of the results are associated to Ramanujan’s...     

An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials

In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an arithmetic formula for these coefficients using the “explicit formula” of prime number theory. In this paper, the author obtains an arithmetic formula for corresponding coefficients associated with the Euler product of Hecke polynomials, which is essentially a product of L-functions attached to weight 2 cusp forms (both newforms and oldforms) over Hecke congruence subgroups Γ₀(N). The nonnegativity of these coefficients gives a criterion for the Riemann hypothesis for all these L-functions at once.     

Non-commutative digit expansions for arithmetic on supersingular elliptic curves

We study non-commutative digit expansions in quaternion rings that arise in the context of supersingular elliptic curves. These digit expansions can be used in a \({\tau}\)-and-add method to speed up arithmetic (scalar multiplication and pairing) on certain families of supersingular elliptic curves in characteristic \({p \geqq 5}\). The basis \({\tau}\) is a quadratic algebraic integer that represents the Frobenius endomorphism of the curve, which is a very fast operation to evaluate. We prove the existence of a finite expansion for every element of the quaternion ring, as well as the equivalence between right and left digit expansions (i.e. the basis \({\tau}\) is placed right, resp. left, to the digit); this expansion turns out to be a non-adjacent form (NAF) for integers, i.e. in every two consecutive digits there is at least one 0.     

A reciprocity theorem for certain q-series found in Ramanujan’s lost notebook

Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s ₁ψ₁ summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D15 B. C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency. A. J. Yee: Research partially supported by a grant from The Number Theory Foundation.     

Riemann-Hilbert theory for problems with vanishing coefficients that arise in nonlinear hydrodynamics

Motivated by a question from mathematical hydrodynamics, this paper studies the solution set of Riemann-Hilbert problems on the unit disc D in of the form

     

The closures of arithmetic progressions in the common division topology on the set of positive integers

In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a factor of a, then it is also a factor of b. The topology T is called the common division topology.     

A linearization for operator polynomials with coefficients in certain operator ideals

Summary For an operator polynomial with coefficients in certain Von Neumann-Schatten classes a linearization is constructed. The linearization also belongs to a Von Neumann-Schatten class, and its regularized Fredholm determinant determines the spectral properties of the original polynomial.     

Inequalities for bounds on effective transport coefficients of two-phase media from power expansions at real points

The effective transport coefficient q of two-phase media is uniquely expressed by a special Stieltjes function f. In this paper, we incorporate into bounds on q, the power expansions of f available at a number of real points. To this end we apply the S-continued fraction technique developed in Baker (Essentials of Padé Approximants, Academic Press, New York, 1975) and Tokarzewski (IFTR Rep 4:3–171, 2005). Moreover, we establish the fundamental inequalities for S-bounds on q. As an example of an application the sequences of upper and lower S-bounds on the effective conductivity of a face-centered cubic lattice of spheres are evaluated.