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Period polynomials, derivatives of L-functions, and zerosof polynomials (2018)
Journal Article
Diamantis, N., & Rolen, L. (2018). Period polynomials, derivatives of L-functions, and zerosof polynomials. Research in the Mathematical Sciences, 5, Article 9. https://doi.org/10.1007/s40687-018-0126-4

Period polynomials have long been fruitful tools for the study of values of L-functions in the context of major outstanding conjectures. In this paper, we survey some facets of this study from the perspective of Eichler cohomology. We discuss ways to... Read More about Period polynomials, derivatives of L-functions, and zerosof polynomials.

Regularized inner products and errors of modularity (2016)
Journal Article
Bringmann, K., Diamantis, N., & Ehlen, S. (2017). Regularized inner products and errors of modularity. International Mathematics Research Notices, 2017(24), 7420-7458. https://doi.org/10.1093/imrn/rnw225

© The Author(s) 2016. We develop a regularization for Petersson inner products of arbitrary weakly holomorphic modular forms, generalizing several known regularizations. As one application, we extend work of Duke, Imamoglu, and Toth on regularized in... Read More about Regularized inner products and errors of modularity.

Fourier coefficients of Eisenstein series formed with modular symbols and their spectral decomposition (2016)
Journal Article
Bruggeman, R., & Diamantis, N. (2016). Fourier coefficients of Eisenstein series formed with modular symbols and their spectral decomposition. Journal of Number Theory, 167, https://doi.org/10.1016/j.jnt.2016.03.009

The Fourier coefficient of a second order Eisenstein series is described as a shifted convolution sum. This description is used to obtain the spectral decomposition of and estimates for the shifted convolution sum.

A correspondence of modular forms and applications to values of L-series (2015)
Journal Article
Diamantis, N., Neururer, M., & Strömberg, F. (in press). A correspondence of modular forms and applications to values of L-series. Research in Number Theory, 1(27), https://doi.org/10.1007/s40993-015-0029-z

An interpretation of the Rogers–Zudilin approach to the Boyd conjectures is established. This is based on a correspondence of modular forms which is of independent interest. We use the reinterpretation for two applications to values of L-series and v... Read More about A correspondence of modular forms and applications to values of L-series.

A converse theorem for double Dirichlet series and Shintani zeta functions (2014)
Journal Article
Diamantis, N., & Goldfeld, D. (2014). A converse theorem for double Dirichlet series and Shintani zeta functions. https://doi.org/10.2969/jmsj/06620449

The main aim of this paper is to obtain a converse theorem for double Dirichlet series and use it to show that the Shintani zeta functions which arise in the theory of prehomogeneous vector spaces are actually linear combinations of Mellin transforms... Read More about A converse theorem for double Dirichlet series and Shintani zeta functions.

Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions (2012)
Journal Article
Bringmann, K., Diamantis, N., & Raum, M. (2013). Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions. Advances in Mathematics, 233(1), 115-134. https://doi.org/10.1016/j.aim.2012.09.025

We introduce a new technique of completion for 1-cohomology which parallels the corresponding technique in the theory of mock modular forms. This technique is applied in the context of non-critical values of L-functions of GL(2) cusp forms. We prove... Read More about Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions.

Second-Order Modular Forms with Characters (2012)
Book Chapter
Blann, T., & Diamantis, N. (2012). Second-Order Modular Forms with Characters. In Fourier Analysis and Number Theory to Radon Transforms and Geometry (55-66). Springer. https://doi.org/10.1007/978-1-4614-4075-8_4

We introduce spaces of second-order modular forms for which the relevant action involves characters. We compute the dimensions of these spaces by constructing explicit bases.

A converse theorem for double Dirichlet series (2011)
Journal Article
Diamantis, N., & Goldfeld, D. (2011). A converse theorem for double Dirichlet series. American Journal of Mathematics, 133(4), 913-938. https://doi.org/10.1353/ajm.2011.0024

We prove that a certain vector valued double Dirichlet series satisfying appropriate functional equations is a Mellin transform of a vector valued metaplectic Eisenstein series. We establish an analogous result for scalar double Dirichlet series of t... Read More about A converse theorem for double Dirichlet series.

New percolation crossing formulas and second-order modular forms (2009)
Journal Article
Diamantis, N., & Kleban, P. (2009). New percolation crossing formulas and second-order modular forms. Communications in Number Theory and Physics, 3(4), 677–696. https://doi.org/10.4310/cntp.2009.v3.n4.a4

We consider the three crossing probability densities for percolation recently found via conformal field theory [23]. We prove that all three of them (i) may be simply expressed in terms of Cardy’s [4] and Watts’ [24] crossing probabilities, (ii) are... Read More about New percolation crossing formulas and second-order modular forms.