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Mathematische Zeitschrift : Volume 286

Mathematische Zeitschrift : Volume 285

Mathematische Zeitschrift : Volume 284

Mathematische Zeitschrift : Volume 284, Issue 3-4, December 2016

Rigid and Schurian modules over cluster-tilted algebras of tame type

Loci of curves with subcanonical points in low genus

Imaginary cones and limit roots of infinite Coxeter groups

A continuous model for quasinilpotent operators

Riesz transforms through reverse Hölder and Poincaré inequalities

On a secant Dirichlet series and Eichler integrals of Eisenstein series

On (2, 4) complete intersection threefolds that contain an Enriques surface

Spectral analysis of the diffusion operator with random jumps from the boundary

Vahlen groups defined over commutative rings

Polynomial inequalities and universal Taylor series

Asymptotic rank of spaces with bicombings

Scaling of conformal blocks and generalized theta functions over $$\overline{\mathcal {M}}_{g,n}$$

Type A images of Galois representations and maximality

Nonintegrability of Hamiltonian system perturbed from integrable system with two singular points

Convergence of the solutions of the discounted equation: the discrete case

Large $${\varvec{p}}^{\prime }$$ -orbits for $${\varvec{p}}$$ -nilpotent linear groups

Slow escaping points of quasiregular mappings

The geometry of Euclidean surfaces with conical singularities

Holomorphic differentials and Laguerre deformation of surfaces

Spectral synthesis for coadjoint orbits of nilpotent Lie groups

Free group representations from vector-valued multiplicative functions, II

On $$L^\infty $$ -BMO estimates for derivatives of the Stokes semigroup

Generic boundary behaviour of Taylor series in Hardy and Bergman spaces

A trace formula for non-unitary representations of a uniform lattice

On $$A_p$$ – $$A_\infty $$ type estimates for square functions

Gevrey smooth topology is proper to detect normalization under Siegel type small divisor conditions

Küchle fivefolds of type c5

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On a secant Dirichlet series and Eichler integrals of Eisenstein series

Content Provider	Springer Nature Link
Author	Straub, Armin Berndt, Bruce C.
Copyright Year	2016
Abstract	We consider the secant Dirichlet series $$\psi _s (\tau ) = \sum _{n = 1}^{\infty } \frac{\sec (\pi n \tau )}{n^s}$$ , recently introduced and studied by Lalín, Rodrigue and Rogers. In particular, we show, as conjectured and partially proven by Lalín, Rodrigue and Rogers, that the values $$\psi _{2 m} (\sqrt{r})$$ , with $$r > 0$$ rational, are rational multiples of $$\pi ^{2 m}$$ . We then put the properties of the secant Dirichlet series into context by showing that, for even s, they are Eichler integrals of odd weight Eisenstein series of level 4. This leads us to consider Eichler integrals of general Eisenstein series and to determine their period polynomials. In the level 1 case, these polynomials were recently shown by Murty, Smyth and Wang to have most of their roots on the unit circle. We provide evidence that this phenomenon extends to the higher level case. This observation complements recent results by Conrey, Farmer and Imamoglu as well as El-Guindy and Raji on zeros of period polynomials of Hecke eigenforms in the level 1 case. Finally, we briefly revisit results of a similar type in the works of Ramanujan.
Ending Page	852
Page Count	26
Starting Page	827
File Format	PDF
ISSN	00255874
e-ISSN	14321823
Journal	Mathematische Zeitschrift
Issue Number	3-4
Volume Number	284
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-05-12
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Trigonometric and exponential sums, general Eichler integrals Holomorphic modular forms of integral weight Ramanujan polynomials Eisenstein series Trigonometric Dirichlet series Mathematics Higher logarithm functions Other functions defined by series and integrals Unimodular polynomials
Content Type	Text
Resource Type	Article
Subject	Mathematics

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