Related papers: Low-lying zeros of quadratic Dirichlet $L$-functio…

We present some monotonicity results for Dirichlet $L$-functions associated to real primitive characters. We show in particular that these Dirichlet $L$-functions are far from being logarithmically completely monotonic. Also, we show that,…

We establish lower bounds for the $2k$-th moment of families of quadratic Dirichlet $L$-functions at the central point for all real $k<0$, assuming a conjecture of S. Chowla on the non-vanishing of these $L$-values.

We extend to Dirichlet L-functions associated with arbitrary primitive characters a range of objects and properties -- including Eisenstein series and period functions -- that were originally introduced and studied by Lewis and Zagier…

Katz and Sarnak conjectured that the statistics of low-lying zeros of various family of $L$-functions matched with the scaling limit of eigenvalues from the random matrix theory. In this paper we confirm this statistic for a family of…

In this report, we present a proof of Levinson's theorem, following the ideas of Matthew P. Young in 2010, which states that one-third of the non-trivial zeros of the Riemann zeta function lie on the critical line, i.e. the line Re(s) =…

Let $\mathcal{L}(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ be an $L$-function in the Selberg class, and $q_{\mathcal{L}}$ its conductor. Let $\ell_0(\mathcal{L})$ be the constant term of the Laurent expansion of $\mathcal{L}'/\mathcal{L}$ at…

We deal with negative square moments of Dirichlet $L$-functions. Summing over characters modulo $q$, we obtain an asymptotic formula for the negative second moment of $L(1,\chi)$ involving conductors. As an application, we give the improved…

For each primitive Dirichlet character $\chi$, a hypothesis ${\rm GRH}^\dagger[\chi]$ is formulated in terms of zeros of the associated $L$-function $L(s,\chi)$. It is shown that for any such character, ${\rm GRH}^\dagger[\chi]$ is…

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…

We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a real, non-principal character modulo $q$. Using Pintz's refinement of Page's theorem, we prove that for $q\geq 3$ the function $L(s, \chi)$ has at most one real zero $\beta$…

We prove some new log-free density theorems for zeros of Dirichlet L-functions (which accordingly are more sharp than earlier ones near to the boundary line of the critical strip). The results can be applied in several problems of prime…

We prove an explicit upper bound of the function $S(t,\chi)$, defined by the argument of Dirichlet $L$-functions. An explicit upper bound of the function $S_1(t)$, defined by the integral of the argument of the Riemann zeta-function, have…

We prove a Lindel\"of-on-average upper bound for the fourth moment of Dirichlet $L$-functions of conductor $q$ along a coset of the subgroup of characters modulo $d$ when $q^*|d$, where $q^*$ is the least positive integer such that…

We establish upper bounds for shifted moments of modular $L$-functions to a fixed modulus as well as quadratic twists of modular $L$-functions under the generalized Riemann hypothesis. Our results are then used to establish bounds for…

The Katz-Sarnak Density Conjecture states that the behavior of zeros of a family of $L$-functions near the central point (as the conductors tend to zero) agree with the behavior of eigenvalues near 1 of a classical compact group (as the…

The moments of quadratic Dirichlet $L$-functions over function fields have recently attracted much attention with the work of Andrade and Keating. In this article, we establish lower bounds for the mean values of the product of quadratic…

The Riemann Zeta-Function is the most studied L-function; it's zeroes give information about the prime numbers. We can associate L-functions to a wide array of objects, and in general, the zeroes of these L-functions give information about…

We evaluate the first moment of the family of primitive quadratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series under the Riemann hypothesis and the Lindel\"of hypothesis. We obtain asymptotic…

We find an explicit upper bound for general $L$-functions on the critical line, assuming the Generalized Riemann Hypothesis, and give as illustrative examples its application to some families of $L$-functions and Dedekind zeta functions.…

An important function attached to a complex simple Lie group $G$ is its asymptotic character $X(\lambda,x)$ (where $\lambda,x$ are real (co)weights of $G$) - the Fourier transform in $x$ of its Duistermaat-Heckman function $DH_\lambda(p)$…