This paper contains estimates of several exponential sums involving Fourier coefficients of certain modular forms. Although the questions which we consider date back, in some cases, to the twenties, our motivation is modern. All of these sums are connected with one approach to the problem of calculating the norm of the Poincaré 0-operator. In this section we state our results (and their antecedents) and then describe their relevance to the θ-operator problem. For simplicity we state and prove all our results for the modular cusp form of weight six, the famous discriminant function

The Fourier transform f of a function is defind by

t ∊ ℝn. A(ℝn) is the isometric image of L1(ℝn) under the Fourier transformation. We extend the Fourier transformation to a mapping of ℒ(ℝn) to itself and denote by A′(ℝn) the isometric image of L∞(ℝn).

Let f be a positive-definite quadratic form with integer coefficients, and denote by c(f) (≥ 1) the class-number of f, that is, the number of classes in the genus of f. I showed in [4] that c(f) ≥ 2 for every f in n ≥ 11 variables; the transformations of [3] were used to make the problem easier. I have since sought to find all the one-class n-ary genera with 3 ≤ n ≤ 10 (the case n = 1 is trivial, and n = 2 is very difficult).

The cosine transforms of functions on the unit sphere play an important role in convex geometry, Banach space theory, stochastic geometry and other areas. Their higher-rank generalization to Grassmann manifolds represents an interesting mathematical object useful for applications. More general integral transforms are introduced that reveal distinctive features of higher-rank objects in full generality. These new transforms are called the composite cosine transforms, by taking into account that their kernels agree with the composite power function of the cone of positive definite symmetric matrices. It is shown that injectivity of the composite cosine transforms can be studied using standard tools of the Fourier analysis on matrix spaces. In the framework of this approach, associated generalized zeta integrals are introduced and new simple proofs given to the relevant functional relations. The technique is based on application of the higher-rank Radon transform on matrix spaces.

As usual, if π is a finite group and ℳ ⊆ ℚπ a maximal order containing , we set

This is well known to be a finite group, and independent of the choice of ℳ.

It has been pointed out to us by Professor L. Schoenfeld that there is a fallacy in the proof of Theorem 3 of our paper “The values of a trigonometrical polynomial at well spaced points” [ Mathematika, 13 (1966), 91–96]. The fallacy occurs in the appeal to Theorem 1 at the end of the proof. If this is to be made explicitly, we must not only put n = dn′ but also put q = dq′ but then the sum over m goes from 1 to q instead of from 1 to q′, and if one allows for this the final result becomes much weakened.

We determine infinite products in the field of Laurent series with the property that the truncations of the product yield every second continued fraction convergent of the product. We mention some related examples and specialize to obtain numerical results.

For each integer n ≥ 2, let β(n) be the sum of the distinct prime divisors of n and let (x) stand for the set of composite integers n ≤ x such that n is a multiple of β(n). Upper and lower bounds are obtained for the cardinality of (x).

In this paper we give alternative proofs of some formulas of Shintani [1, 2] and Zagier [3] concerning Kronecker limit formulas for real quadratic fields.

Two basic approaches have been used to develop explicit formulae for the number of classes in a genus of binary quadratic lattices over an algebraic number field. Analytic machinery in the form of the Minkowski-Siegel Mass Formula or the Tamagawa number of an algebraic group was employed by Pfeuffer [13] and Shyr [17] to obtain such a formula for maximal positive definite lattices over totally real number fields. On the other hand, Peters [10] observed that a formula applicable to maximal lattices over any number field can be deduced by algebraic methods from the theory of quadratic field extensions. Using group-theoretic techniques set up by the present authors [3] along with the calculation of certain local unit indices, Korner [6] derived the corresponding formula for non-maximal lattices.

Some interlacing properties of the zeros of the generalized Airy functions A1(z, p) are given for non-positive integral values of p. The result that A1 (z,p) has no real zero for is extended to show that all the zeros of A1(z,p) are real and simple if . It is also shown that all the zeros of the functions Bk(z,p, 1) for k = 1, 2, 3 are simple for non-positive integral p.

Suppose that f(z) is non-constant and meromorphic in the plane and that, for some k≥= 1, a0(z),…, ak(z) are meromorphic in the plane with

for j' = 0,…, k. Here, using standard notation from [3], S(r,f) denotes any quantity satisfying S(r,f) = o(T(r,f)) as r→ ∞, possibly outside a set of finite linear measure. Then, setting

we have ([3, p. 57])

Theorem A. Suppose that f(z) is non-constant and meromorphic in the plane, and thatψ (z) given by (1.2) and (1.1) and is non-constant. Then

where N0(r, l/ψ') counts only zeros of ψ' which are not zeros of ψ − 1, and thecounting functions count points without regard to multiplicity.

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain Ω in ℝN. We consider deformations ϕ(Ω) of Ω obtained by means of a locally Lipschitz homeomorphism ϕ and we estimate the variation of the eigenfunctions and eigenvalues upon variation of ϕ. We prove general stability estimates without assuming uniform upper bounds for the gradients of the maps ϕ. As an application, we obtain estimates on the rate of convergence for eigenvalues and eigenfunctions when a domain with an outward cusp is approximated by a sequence of Lipschitz domains.

Generalizing a result (the case $k= 1$) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension $2k+ 1$ belongs to the generalized Walkup class ${ \mathcal{K} }_{k} (2k+ 1)$, i.e., all its vertex links are $k$-stacked spheres. This is surprising since it is far from obvious that the vertex links of polytopal upper bound spheres should have any special combinatorial structure. It has been conjectured that for $d\not = 2k+ 1$, all $(k+ 1)$-neighborly members of the class ${ \mathcal{K} }_{k} (d)$ are tight. The result of this paper shows that the hypothesis $d\not = 2k+ 1$ is essential for every value of $k\geq 1$.

We obtain the formula for the twisted harmonic second moment of the L-functions associated with primitive Hecke eigenforms of weight 2. A consequence of our mean-value theorem is reminiscent of recent results of Conrey and Young on the reciprocity formula for the twisted second moment of Dirichlet L-functions.

For any given integer $k\geqslant 2$ we prove the existence of infinitely many $q$ and characters $\unicode[STIX]{x1D712}\,(\text{mod}\;q)$ of order $k$ such that $|L(1,\unicode[STIX]{x1D712})|\geqslant (\text{e}^{\unicode[STIX]{x1D6FE}}+o(1))\log \log q$. We believe this bound to be the best possible. When the order $k$ is even, we obtain similar results for $L(1,\unicode[STIX]{x1D712})$ and $L(1,\unicode[STIX]{x1D712}\unicode[STIX]{x1D709})$, where $\unicode[STIX]{x1D712}$ is restricted to even (or odd) characters of order $k$ and $\unicode[STIX]{x1D709}$ is a fixed quadratic character. As an application of these results, we exhibit large even-order character sums, which are likely to be optimal.

In a recent important paper, Hoffstein and Hulse [Multiple Dirichlet series and shifted convolutions, arXiv:1110.4868v2] generalized the notion of Rankin–Selberg convolution $L$-functions by defining shifted convolution$L$-functions. We investigate symmetrized versions of their functions, and we prove that the generating functions of certain special values are linear combinations of weakly holomorphic quasimodular forms and “mixed mock modular” forms.

Let $\unicode[STIX]{x1D6E4}\subseteq \operatorname{PSL}(2,\mathbf{R})$ be a finite-volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\unicode[STIX]{x1D6E4}$-orbit of $z$ in a hyperbolic circle around $w$ of radius $R$, where $z$ and $w$ are given points of the upper half plane and $R$ is a large number. An estimate with error term $\text{e}^{(2/3)R}$ is known, and this has not been improved for any group. Recently, Risager and Petridis proved that in the special case $\unicode[STIX]{x1D6E4}=\operatorname{PSL}(2,\mathbf{Z})$ taking $z=w$ and averaging over $z$ in a certain way the error term can be improved to $\text{e}^{(7/12+\unicode[STIX]{x1D716})R}$. Here we show such an improvement for a general $\unicode[STIX]{x1D6E4}$; our error term is $\text{e}^{(5/8+\unicode[STIX]{x1D716})R}$ (which is better than $\text{e}^{(2/3)R}$ but weaker than the estimate of Risager and Petridis in the case $\unicode[STIX]{x1D6E4}=\operatorname{PSL}(2,\mathbf{Z})$). Our main tool is our generalization of the Selberg trace formula proved earlier.

We examine the minimal magnitude of perturbations necessary to change the number $N$ of static equilibrium points of a convex solid $K$. We call the normalized volume of the minimally necessary truncation robustness and we seek shapes with maximal robustness for fixed values of $N$. While the upward robustness (referring to the increase of $N$) of smooth, homogeneous convex solids is known to be zero, little is known about their downward robustness. The difficulty of the latter problem is related to the coupling (via integrals) between the geometry of the hull $\mathrm{bd} \hspace{0.167em} K$ and the location of the center of gravity $G$. Here we first investigate two simpler, decoupled problems by examining truncations of $\mathrm{bd} \hspace{0.167em} K$ with $G$ fixed, and displacements of $G$ with $\mathrm{bd} \hspace{0.167em} K$ fixed, leading to the concept of external and internal robustness, respectively. In dimension 2, we find that for any fixed number $N= 2S$, the convex solids with both maximal external and maximal internal robustness are regular $S$-gons. Based on this result we conjecture that regular polygons have maximal downward robustness also in the original, coupled problem. We also show that in the decoupled problems, three-dimensional regular polyhedra have maximal internal robustness, however, only under additional constraints. Finally, we prove results for the full problem in the case of three-dimensional solids. These results appear to explain why monostatic pebbles (with either one stable or one unstable point of equilibrium) are found so rarely in nature.

Introduced in Schmidt and Summerer [Parametric geometry of numbers and applications. Acta Arith.140 (2009), 67–91], approximation exponents $\text{}\underline{\unicode[STIX]{x1D711}}_{i},\overline{\unicode[STIX]{x1D711}}_{i}$, $(i=1,2,3)$, attached to points $\boldsymbol{\unicode[STIX]{x1D709}}=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2})$ in $\mathbb{R}^{2}$, give information on Diophantine approximation properties of these points. Laurent [Exponents of Diophantine approximation in dimension two. Canad. J. Math.61 (2009), 165–189] had described all possible quadruples $(\text{}\underline{\unicode[STIX]{x1D711}}_{1},\overline{\unicode[STIX]{x1D711}}_{1},\text{}\underline{\unicode[STIX]{x1D711}}_{3},\overline{\unicode[STIX]{x1D711}}_{3})$ arising in this way. Our emphasis here will be on $\text{}\underline{\unicode[STIX]{x1D711}}_{2},\overline{\unicode[STIX]{x1D711}}_{2}$ and the construction of suitable “$3$-systems”.