Spaces of modular forms generated by eta-quotients

In this paper, we study the spaces of modular forms on Γ₀(N) generated by eta-quotients, where the genus of Γ₀(N) is zero or N is a prime. Our results give an answer to a question of Ono (2004, Problem 1.68). 2000 Mathematics Subject Classification Primary—11F20, 11F11     

An inner product relation on Saito-Kurokawa lifts

Let f be a newform of weight 2k−2 and level M with M an odd square-free integer. Via the Saito-Kurokawa correspondence there is associated to f a Siegel newform F _f of weight k and level M. In this paper we provide a formula relating the Petersson products {F _f , F _f } and {f, f}. We use this result to give a new proof of a special case of a well-known result of Shimura on the algebraicity of a special value of a Rankin convolution L-function. 2000 Mathematics Subject ClassificationPrimary—11F32; Secondary—11F46, 11F67     

Borwein's conjecture on average over arithmetic progressions

We provide asymptotic formulas for sums over arithmetic progressions of coefficients of products of the form

On the verification of Clark's example of a euclidean but not norm-euclidean number field

In the preceding paper [2], D. Clark proved—modulo a finite amount of computation—that the ring of integersR of admits explicit euclidean algorithms, although it is not euclidean for the norm: In fact, every completely multiplicative function ϕ:R→R _>-0 which sends the prime elements above 23 to a value larger than 25 and which agrees with the absolute norm at all other primes defines a euclidean algorithm forR. The referee had felt that an independent verification of the computer-assisted proofs of Lemmas 1 and 2 of [2] was desirable, and that it should be carried out separately from the refereeing process in the light of the public, conforming to C. Lam's eloquent suggestions [3]. F. Lemmermeyer and the present author succeeded in confirming Clark's result (independently of each other). This note gives some details of the methods employed in the verifications.     

The irrationality of a number theoretical series

Denote by σ _k (n) the sum of the k-th powers of the divisors of n, and let . We prove that Schinzel’s conjecture H implies that S _k is irrational, and give an unconditional proof for the case k = 3. 2000 Mathematics Subject Classification Primary—11A25, 11N36, 11J72     

On completely multiplicative functions whose values are roots of unity

Summary We prove that if a complex valued completely multiplicative function F and a positive integer ℓ≦5 satisfy the condition F(N) = U_ℓ, where U_ℓ is the set of all ℓ-th roots of unity, then {F(n+1) F(n) ∣ nε N} = U_ℓ.     

Computing the Ramanujan tau function

We show that the Ramanujan tau function τ(n) can be computed by a randomized algorithm that runs in time for every O( ) assuming the Generalized Riemann Hypothesis. The same method also yields a deterministic algorithm that runs in time O( ) (without any assumptions) for every ε > 0 to compute τ(n). Previous algorithms to compute τ(n) require Ω(n) time. Research supported in part by NSF grant CCR-9988202 2000 Mathematics Subject Classification Primary—11-04; Secondary—11Y55, 11F11     

Asymptotic Expansions in Non-central Limit Theorems for Quadratic Forms

We consider quadratic forms of the type

Cubic elliptic functions

The function

Quadratic transformations for a function that generalizes 2F1 and the Askey-Wilson polynomials

In previous papers we introduced and studied a ‘relativistic’ hypergeometric function R(a ₊, a ₋, c; v, ) that satisfies four hyperbolic difference equations of Askey-Wilson type. Specializing the family of couplings c∊ to suitable two-dimensional subfamilies, we obtain doubling identities that may be viewed as generalized quadratic transformations. Specifically, they give rise to a quadratic transformation for ₂ F ₁ in the ‘nonrelativistic’ limit, and they yield quadratic transformations for the Askey-Wilson polynomials when the variables v or are suitably discretized. For the general coupling case, we also study the bearing of several previous results on the Askey-Wilson polynomials. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D45, 39A70     

Number of solutions for cubic Thue equations with automorphisms

Let be an irreducible cubic form with positive discriminant, and with non-trivial automorphisms. We show that the Thue equation F(x,y) = 1 has at most three integer solutions except for a few known cases. For the proof, we use an explicitly expressed cubic form which is equivalent to F. To obtain an upper bound for the size of solutions, we use the Padé approximation method developed in our former work. To obtain a lower bound for the size of solutions, we use a result of R. Okazaki on gaps between solutions, which is obtained by geometric consideration. 2000 Mathematics Subject Classification Primary—11D25, 11D59     

Threshold for the volume spanned by random points with independent coordinates

Let μ be an even compactly supported Borel probability measure on the real line. For every N > n consider N independent random vectors X ₁, ..., X _N in ℝ ⁿ , with independent coordinates having distribution μ. We establish a sharp threshold for the volume of the random polytope K _N ≔ conv{X ₁, ..., X _N }, provided that the Legendre transform λ of the cumulant generating function of μ satisfies the condition

On some properties of polynomials related to hypergeometric modular forms

We find the discriminants, Galois groups, and prove the irreducibility of certain hypergeometric polynomials, which are closely related to modular forms and supersingular elliptic curves. 2000 Mathematics Subject Classification Primary—33C45; Secondary—11F11     

Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals

 If u is a minimizer of ∫_Ω F(|∇u|)dx−∫_Ω udμ, then the pointwise estimate

New transformations for elliptic hypergeometric series on the root system A n

Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system A _n , with different dimensions n. We give, with a new, elementary proof, an elliptic extension of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series. 2000 Mathematics Subject Classification Primary—33D67; Secondary—11F50     

Generalized bisymmetry on a restricted domain

Let X ⊂ ℝ be an interval of positive length and define the set Δ = {(x, y) ∈ X × X | x ≧ y}. We give the solution of the equation

On nonatomic submeasures on {\mathbb{N}}

A submeasure μ defined on the subsets of is nonatomic if for every ℓ ≥ 1 there exists a partition of into a finite number of parts on which μ is bounded from above by 1/ℓ. In this paper we answer several natural questions concerning nonatomic submeasures d _F that are determined (like the standard density) by a family F of finite subsets of . We first show that if the number of n-element sets in F grows at most exponentially with n, then d _F is nonatomic; but if this growth condition fails, then d _F need not be nonatomic in general. We next prove that, for a nonatomic submeasure d _F , the minimal number of sets in a 1/ℓ-small partition of can grow arbitrarily fast with ℓ. We also give a simple example of a nonatomic submeasure that is not equivalent to a submeasure of type d _F . The second author acknowledges a generous support of the Foundation for Polish Science.     

The Dimension of Subsets of Moran Sets Determined by the Success Run Behaviour of Their Codings

 Let be a Moran set associated with the set . Let Γ be a non-empty subset of with non-empty complement. Associated with the behaviour of success run of symbols from Γ in the coding space is a decomposition of F such that

Splitting metaplectic covers of dual reductive pairs

The symplectic group Sp(N, F) over a local fieldF (other than ℂ) has a unique non-trivial twofold central extension. The inclusion of {±1} into the circle ℂ¹ induces an extension . In this paper, an explicit splitting of the restriction of this extension to a dual reductive pair (G, H) in Sp(n, F) is given in all cases in which it exists. Such an explicit splitting is often an essential technical ingredient in the study of the local theta correspondence for the dual pair [4]. Partially supported by NSF Grant DMS-9003109.