As usual, define Dedekind's eta-function η(z) by the infinite product

$$ \eta(z):=q^{1/24}\prod_{n=1}^{\infty} \big(1-q^n\big) \quad \big(q:=e^{2\pi i z} \text{ throughout}\big). $$

In a recent paper, D. Zagier proved that (note: empty products equal 1 throughout)

$$ \sum_{n=0}^{\infty} \Big(\eta(24z)-q\big(1-q^{24}\big) \big(1-q^{48}\big)\cdots \big(1-q^{24n}\big)\Big) =\eta(24z)D(q)+E(q), $$

where the series D(q) and E(q) are defined by

$\begin{align*} D(q)&=-\frac{1}{2}+\sum_{n=1}^{\infty}\frac{q^{24n}}{1-q^{24n}}\\ &=-\frac{1}{2}+\sum_{n=1}^{\infty}d(n)q^{24n}\\ &=-\frac{1}{2}+q^{24}+2q^{48}+2q^{72}+3q^{96}+\cdots\quad,\\ E(q)&=\frac{1}{2}\sum_{n=1}^{\infty} \bigg(\frac{12}{n}\bigg)nq^{n^2} =\frac{1}{2}q-\frac{5}{2}q^{25}- \frac{7}{2}q^{49}+\frac{11}{2}q^{121}+\cdots\quad. \end{align*}$

Here d(n) denotes the number of positive divisors of n. We obtain two infinite families of such identities and describe some consequences for L-functions and partitions. For example, if χ₂ is the Kronecker character for ℚ($\sqrt{2}$), these identities can be used to show that

$\begin{align*} &-2e^{-t/8}\sum_{n=0}^{\infty} \frac{\big(1-e^{-2t}\big)\big(1-e^{-4t}\big) \cdots\big(1-e^{-2nt}\big)} {\big(1+e^{-t}\big)\big(1+e^{-3t}\big) \cdots\big(1+e^{-(2n+1)t}\big)} \\ &\hspace{110pt}=\sum_{n=0}^{\infty} \bigg(\frac{-1}{8}\bigg)^n\cdot L(\chi_{2},-2n-1)\cdot \frac{t^{n}}{n!}. \end{align*}$

On a bounded pseudoconvex domain Ω in ℂⁿ with a plurisubharmonic Lipschitz defining function, we prove that the $\overline\partial$-Neumann operator is bounded on Sobolev (1/2)-spaces.

We study the concentration properties of high-energy eigenfunctions for the Laplace-Beltrami operator of the hyperbolic plane with special consideration of automorphic eigenfunctions. At the center of our investigation is the microlocal lift of an eigenfunction to SL(2;ℝ) introduced by S. Zelditch. The microlocal lift is based on S. Helgason's Fourier transform and has a straightforward description in terms of the Lie algebra sl(2;ℝ). We begin with an elementary demonstration of Zelditch's exact differential equation for the microlocal lift. Further, for a sequence of suitably bounded eigenfunctions with eigenvalues tending to infinity, we show that the limit of microlocal lifts is a geodesic-flow-invariant positive measure.

Our main consideration is the microlocal lifts of the elementary eigenfunctions constructed from the Macdonald-Bessel functions. We find that with a scaling of auxiliary parameters the corresponding high-energy limit converges to the positive Dirac measure for the lift to SL(2;ℝ) of a single geodesic on the upper half-plane. In particular, at high energy the microlocal lift of a Macdonald-Bessel function is concentrated along the lift of a single geodesic. We further find that the convergence is uniform in parameters, and thus our considerations can be applied to study automorphic eigenfunctions, in particular, the Maass cusp forms for certain cofinite subgroups of SL(2;ℝ). A formula is presented for the microlocal lift of a cusp form as an integral of a family of Dirac measures and a measure given directly by sums of products of the classical Fourier coefficients of the cusp form. The formula establishes that questions on "quantum chaos" for automorphic eigenfunctions are equivalent to classically stated questions on twisted sums of coefficients. In particular, we find that the uniform distribution for the microlocal limit is equivalent to a uniform distribution for the limit of twisted coefficient sums. We extend the considerations to the case of the nonholomorphic Eisenstein series for the modular group. We find that a result of W. Luo and P. Sarnak is equivalent to a limit-sum formula involving the Riemann zeta-function and the elementary divisor function. The formula is suggestive of a formula of S. Ramanujan.

We introduce and describe one-dimensional cyclotomic Gauss-Selberg sums generalizing the classical Gaussian sums. They correspond to irreducible self-dual unitary spherical representations of the one-dimensional double affine Hecke algebra.

Let Ω⊂ℝⁿ be a bounded, convex, and open set with real analytic boundary. Let T_Ω⊂ℂⁿ be the tube with base Ω, and let $\mathcal{B}$ be the Bergman kernel of T_Ω. If Ω is strongly convex, then $\mathcal{B}$ is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. In this situation we relate the off-diagonal points where analyticity fails to the characteristic lines. These lines are contained in the boundary of T_Ω, and they are projections of the Treves curves. These curves are symplectic invariants that are determined by the CR (Cauchy-Riemann) structure of the boundary of T_Ω. Note that Treves curves exist only when Ω has at least one weakly convex boundary point.

We consider the Weil-Petersson form on the Teichmüller space $\mathscr {T}$(S) of a surface S of genus at least 2, and we compute it in terms of the shearing coordinates for $\mathscr {T}$(S) associated to a geodesic lamination λ on S. In the corresponding expression, the Weil-Petersson form coincides with Thurston's intersection form on the space of transverse cocycles for λ.

We consider the Cauchy data associated to the Schrödinger equation with a potential on a bounded domain Ω⊂ℝⁿ, n≥3. We show that the integral of the potential over a two-plane Π is determined by the Cauchy data of certain exponentially growing solutions on any open subset $\mathscr{U}$⊂∂Ω which contains Π∩∂Ω.