Related papers: Value-distribution of cubic Hecke $L$-functions

Set $K=\mathbb{Q}(i)$ and suppose that $c\in \mathbb{Z}[i]$ is a square-free algebraic integer with $c\equiv 1 \imod{\langle16\rangle}$. Let $L(s,\chi_{c})$ denote the Hecke $L$-function associated with the quartic residue character modulo…

We establish the two-dimensional asymptotic distributions of the logarithm and logarithmic derivative of $L$-functions associated with a family of cubic Hecke characters. A crucial ingredient in the proof of our main result is an…

We study the distributions of values of the logarithmic derivatives of the Dedekind zeta functions on a fixed vertical line. The main object is determining and investigating the density functions of such value-distributions for any…

We establish an unconditional asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For $\Re(s)=\sigma$ satisfying $(\log T)^{-1/3+\epsilon} \leq (2\sigma-1) \leq (\log \log…

Arising from the factorizations of Dedekind zeta-functions of cubic fields, we obtain Artin $L$-functions of certain two-dimensional representations. In this paper, we study the value-distribution of such Artin $L$-functions for families of…

We formulate an abstract notion of equidistribution for families of $\lambda$-probability spaces parameterized by admissible $\mathbb{Z}$-sets. Under the assumption of equidistribution, we show that the $\sigma$-moment generating functions…

Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive…

The main goal of this paper is to prove a formula that expresses the limit behaviour of Dedekind zeta functions for $\Re s > 1/2$ in families of number fields, assuming that the Generalized Riemann Hypothesis holds. This result can be…

We study the distribution of large (and small) values of several families of $L$-functions on a line $\text{Re(s)}=\sigma$ where $1/2<\sigma<1$. We consider the Riemann zeta function $\zeta(s)$ in the $t$-aspect, Dirichlet $L$-functions in…

We consider the logarithm of the central value $\log L(1/2)$ in the orthogonal family ${L(s,f)}_{f \in H_k}$ where $H_k$ is the set of weight $k$ Hecke-eigen cusp form for $SL_2(\mathbb{Z})$, and in the symplectic family…

In this paper we study asymptotic properties of families of zeta and $L$-functions over finite fields. We do it in the context of three main problems: the basic inequality, the Brauer--Siegel type results and the results on distribution of…

Let $\mathbb{K} = \mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic number field of class number $1$ and $\mathcal{O}_{\mathbb{K}}$ its ring of integers. We study a family of Hecke $L$-functions associated to angular characters on the…

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$. Let $F(s)= \sum_{n=1}^\infty f(n)n^{-s}$ be the associated Dirichlet series and $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ be the truncated Dirichlet…

Let $D\equiv 1\bmod{4}$ be a fundamental discriminant of a real quadratic field. We construct an analogue of the classical Dedekind eta function for the Hecke group $H(\sqrt{D})$. This gives rise to a new family of holomorphic modular…

The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta…

Let $\omega$ be a primitive cubic root of unity. We study the non-vanishing problem for the family of Hecke $L$-functions associated to primitive cubic characters defined over the Eisenstein quadratic number field $\mathbb{Q}(\omega)$. We…

Let $\mathcal{O}$ be the ring of integers for some number field $F$. Let $\chi(x)\in \mathcal{O}[x]$ be a regular monic polynomial of degree $n$. We study the asymptotic count of integral $n\times n$ matrices over $\mathcal{O}$ with the…

For a fixed integer $k\ge 3$ and fixed $1/2 < \sigma > 1$ we consider $$ \int_1^T |\zeta(\sigma + it)|^{2k}dt = \sum_{n=1}^\infty d_k^2(n)n^{-2\sigma}T + R(k,\sigma;T), $$ where $R(k,\sigma;T) = o(T) (T\to\infty)$ is the error term in the…

Given a number field $K \neq \mathbb{Q}$, in a now classic work, Stark pinpointed the possible source of a so-called Landau-Siegel zero of the Dedekind zeta function $\zeta_K(s)$ and used this to give effective upper and lower bounds on the…

This is the first installment in a series of papers devoted to examining certain aspects of the asymptotic value distribution and distribution of zeros manifested by members of a broad class of linear combinations of L-functions in the…