Abstract In [Sa], it was proved that the Selberg zeta function for SL2(Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL2(Z). As applications, we study the Brun-Titc...

We consider the t-hook functions on partitions $$f_{a,t}: \mathcal {P}\rightarrow \mathbb {C}$$ defined by $$\begin{aligned} f_{a,t}(\lambda ):=t^{a-1} \sum _{h\in {H}_t(\lambda )}\frac{1}{h^a}, \end{aligned}$$ where $$\mathcal )$$ is multiset of partition hook numbers that are multiples t. The Bloch–Okounkov q-brackets $$\langle f_{a,t}\rangle _q$$ include Eichler integrals classical Eisenstei...

We study eta invariants of Dirac operators and regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner Müller (see [18], [19]) and use relative traces to define the eta function and the zeta function. We show regularities of eta and zeta functions at the origin so that we can define the eta invariant and the regularized determinant. By the Selberg tra...

We introduce and describe one-dimensional cyclotomic Gauss-Selberg sums generalizing the classical Gaussian sums. They correspond to irreducible self-dual unitary spherical representations of the one-dimensional double affine Hecke algebra. Introduction This paper is about applications of the one-dimensional double affine Hecke algebra to q-Gauss integrals and Gaussian sums. The double affine H...

We rewrite Arthur’s asymptotic formula for weighted orbital integrals on real groups with the aid of a residue calculus and extend the resulting formula to the Schwartz space. Then we extract the available information about the coefficients in the decomposition of the Fourier transforms of Arthur’s invariant distributions IM (γ) in terms of standard solutions of the pertinent holonomic system o...

s of papers by Alessandro Zaccagnini Alessandro Zaccagnini Dipartimento di Matematica e Informatica, Università di Parma, Parco Area delle Scienze, 53/A, Campus Universitario 43124 Parma, Italy. [email protected] April 29, 2013 This file contains full references and abstracts of my papers. Chronologically, these deal with the following topics: • Large gaps between primes in arithme...

Mathematisch-naturwissenschaftlichen Fakultät Doctor of Philosophy Twisted Second Moments and Explicit Formulae of the Riemann Zeta-Function by Nicolas Martinez Robles Verschiedene Aspekte, die analytische Zahlentheorie und die Riemann zeta-Funktion verbinden, werden erweitert. Dies beinhaltet: 1. explizite Formeln, die eine Verbindung zwischen der Möbiusfunktion und den nichttrivialen Nullstel...

Introduction 2 1. Preliminaries 6 1.1. Algebraic modular forms 6 1.2. Modular forms over C 9 1.3. p-adic modular forms 11 1.4. Elliptic curves with complex multiplication 12 1.5. Values of modular forms at CM points 14 2. Generalised Heegner cycles 15 2.1. Kuga-Sato varieties 15 2.2. The variety Xr and its cohomology 18 2.3. Definition of the cycles 19 2.4. Relation with Heegner cycles and L-se...