Abstracts for Conference on "Elementare und Analytische Zahlentheorie (ELAZ)", September 3 - 7, 2018

In this talk we will discuss recent developments in the theory of indefinite theta series focusing on examples. This is joint work with Steve Kudla.

In this talk we define a Shintani lift for harmonic Maass forms. We show that it maps weight $2k+2$ harmonic Maass forms to harmonic Maass forms of weight $3/2+k$ and describe its Fourier coefficients in terms of traces of CM values and regularized cycle integrals of the input harmonic Maass forms. Moreover, we present some applications of this construction. If time permits, we will also explain the extension of the lift to meromorphic modular forms. All of this is joint work with Markus Schwagenscheidt and the last part also with Kathrin Bringmann.

Hardy's function is ($t \in \mathbb R$)
$$
Z(t)  := \zeta(1/2+it)\bigl(\chi(1/2+it)\bigr)^{-1/2},
$$
where $\zeta(s) \;=\; \chi(s)\zeta(1-s)$ is the functional equation for the Riemann zeta-function $\zeta(s)$.
The argument function is
$$
S(t) \;:=\; \frac{1}{\pi}\arg \zeta(1/2 + it)\qquad(t>0, \;t \ne \gamma),
$$
where $\rho = \beta+i\gamma$ denotes generic complex zeros of $\zeta(s)$. If $t=\gamma,$
$S(t) \;=\; S(t+0)$. These important functions are real-valued, so one may naturally ask