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

Alternatively have a look at the program.

## Universal quadratic forms over real quadratic fields

A positive integral quadratic form is called universal if it represents all positive integers, such as sums of four squares. This definition makes sense for totally positive quadratic forms over the ring of integers of totally real number fields. For instance, Hans Maa{\ss} showed that sums of three squares are universal over $\mathbb{Q}(\sqrt{5})$.

## Indefinite theta series via incomplete theta integrals

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.

## Arithmetic statistics of modular symbols I

modular symbols in order to understand central values of twists of elliptic curve $L$-functions. Two of

## Arithmetic statistics of modular symbols II

modular symbols in order to understand central values of twists of elliptic curve $L$-functions. Two of

these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols.

## Diophantine inequalities for ternary diagonal forms

We discuss small solutions to ternary diagonal inequalities of any degree where all of the variables are assumed to be of size $P$. We study this problem on average over a one-parameter family of forms and discuss a generalization of work of Bourgain on generic ternary diagonal quadratic forms to higher degree.

## Shintani theta lifts of harmonic Maass forms

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.

## The distribution of Hardy's function $Z(t)$ and the argument function $S(t)$

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

## Analytic techniques in Moonshine

About 20 years after Borcherds's seminal proof of the Conway-Norton's Monstrous Moonshine Conjecture, many other Moonshine phenomena have been discovered and proved, at least abstractly. Most prominent examples include Umbral Moonshine, Thompson Moonshine, and most recently O'Nan Moonshine. In my talk, I present some of the analytic methods that are employed in the proofs of these more recent Moonshine results.

## Divisor-sum fibers

Let $s(\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) =\sum_{d\mid n,~d<n}d$.

Erdös--Granville--Pomerance--Spiro conjectured that, for any set $\mathcal{A}$ of asymptotic

density zero, the preimage set $s^{-1}(\mathcal{A})$ also has density zero. We prove a weak

form of this conjecture. In particular, we show that the EGPS conjecture holds for

infinite sets with counting function $O(x^{\frac12 + \epsilon(x)})$. We also disprove a hypothesis

from the same paper of EGPS by showing that for any positive numbers $\alpha$ and $\epsilon$,

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