A generalization of Knopp’s Observation on Ramanujan’s tau-function

The Ramanujan Journal - We establish a vast generalization of an observation made by Marvin Knopp half a century ago concerning the nonvanishing of Ramanujan’s tau-function.     

On the sign changes of coefficients of general Dirichlet series

Under what conditions do the (possibly complex) coefficients of a general Dirichlet series exhibit oscillatory behavior? In this work we invoke Laguerre's Rule of Signs and Landau's Theorem to provide a rather simple answer to this question. Furthermore, we explain how our result easily applies to a multitude of functions.

     

A generalization of the Lipschitz summation formula and some applications

The Lipschitz formula is extended to a two-variable form. While the theorem itself is of independent interest, we justify its existence further by indicating several applications in the theory of modular forms.

     

On the Signs of Fourier Coefficients of Cusp Forms

Let be a discrete subgroup of SL(2, ) with a fundamental region of finite hyperbolic volume. (Then, is a finitely generated Fuchsian group of the first kind.) Let 0}} {a(n)e^{2\pi i(n + {\kappa })z/{\lambda }} } ,{ }z \in \mathcal{H}.$$ " align="middle" border="0"> be a nontrivial cusp form, with multiplier system, with respect to . Responding to a question of Geoffrey Mason, the authors present simple proofs of the following two results, under natural restrictions upon . Theorem. If the coefficients a(n) are real for all n, then the sequence {a(n)} has infinitely many changes of sign. Theorem. Either the sequence {Re a(n)} has infinitely many sign changes or Re a(n) = 0 for all n. The same holds for the sequence {Im a(n)}.     

On the oscillatory behavior of certain arithmetic functions associated with automorphic forms

We establish the oscillatory behavior of several significant classes of arithmetic functions that arise (at least presumably) in the study of automorphic forms. Specifically, we examine general L-functions conjectured to satisfy the Grand Riemann Hypothesis, Dirichlet series associated with classical entire forms of real weight and multiplier system, Rankin-Selberg convolutions (both “naive” and “modified”), and spinor zeta-functions of Hecke eigenforms on the Siegel modular group of genus two. For the second class we extend results obtained previously and jointly by M. Knopp, W. Kohnen, and the author, whereas for the fourth class we provide a new proof of a relatively recent result of W. Kohnen.     

On Solutions to Riemann's Functional Equation

In 1921 Hamburger proved that Riemann's functional equation characterizes the Riemann zeta function in the space of functions representable by ordinary Dirichlet series satisfying certain regularity conditions. We consider solutions to a more general functional equation with real weight k. In the case of Hamburger's theorem, k = . We show that, under suitable conditions, the generalized functional equation admits no nontrivial solutions for k > 0 unless k = . Our proof generalizes an elegant proof of Hamburger's theorem given by Siegel, and employs a generalized integral transform.1997 Sunrise Way     

The Evolving Etiology of Bilateral Vocal Fold Immobility

In the past, bilateral vocal fold immobility (BVFI) occurred most commonly after thyroidectomy. However, no large series documenting the etiology of adult BVFI has been published within the past fifteen years. This study reviews the etiologic patterns of BVFI at our institutions. We compare BVFI from before and after 1980. We also review combined studies of unilateral vocal fold immobility (UVFI) to compare and unilateral versus bilateral etiologic trends. In comparison with previously published series, fewer cases of BVFI present today as a complication of thyroid surgery and more as the result of malignancies and nonsurgical trauma. Unfortunately, BVFI caused by malignancy is not usually an initial sign of local disease, but an ominous sign of recurrence or metastases. In comparing UVFI and BVFI we found that thyroidectomy causes a higher percentage of BVFI than of UVFI. Over one-third of UVFI cases were caused by neoplasm which further underscores the potential seriousness of immobile vocal folds and the need for careful investigation.