Related papers: Ratios conjecture for primitive quadratic Hecke $L…

For Dirichlet $L$-functions in $\mathbb{F}_q [T]$ we obtain a hybrid Euler-Hadamard product formula. We make a splitting conjecture, namely that the $2k$-th moment of the Dirichlet $L$-functions at $\frac{1}{2}$, averaged over primitive…

Let $f$ be a Hecke-Maass cusp form for the full modular group and let $\chi$ be a primitive Dirichlet character modulo a prime $q$. Let $s_0=\sigma_0+it_0$ with $\frac{1}{2}\leq\sigma_0<1$. We improve the error term for the first moment of…

We use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change L-functions averaged over Hilbert modular forms of a fixed weight and level. We apply these formulas to the subconvexity…

In this paper we obtain a precise formula for the $1$-level density of $L$-functions attached to non-Galois cubic Dedekind zeta functions. We find a secondary term which is unique to this context, in the sense that no lower-order term of…

We establish asymptotic formulae for the first and second moments of quadratic Dirichlet $L$--functions, at the centre of the critical strip, associated to the real quadratic function field $k(\sqrt{P})$ and inert imaginary quadratic…

We establish restrictions on the Hilbert function of standard graded Gorenstein algebras with only quadratic relations. Furthermore, we pose some intriguing conjectures and provide evidence for them by proving them in some cases using a…

Let $\lambda$ be a self-dual Hecke character over an imaginary quadratic field $K$ of infinity type $(1,0)$. Let $\ell$ and $p$ be primes which are coprime to $6N_{K/\mathbb{Q}}({\mathrm cond}(\lambda))$. We determine the $\ell$-adic…

In the study of Dirichlet series with arithmetic significance there has appeared (through the study of known examples) certain expectations, namely (i) if a functional equation and Euler product exists, then it is likely that a type of…

Linearly independent Dirichlet L-functions satisfying the same Riemann-type of functional equation have been supposed for long time to possess off critical line non trivial zeros. We are taking a closer look into this problem and into its…

We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with $L^2$ boundary data. We…

In this paper we apply to the zeros of families of $L$-functions with orthogonal or symplectic symmetry the method that Conrey and Snaith used to calculate the $n$-correlation of the zeros of the Riemann zeta function. This method uses the…

In this article, we study the density conjecture of Katz and Sarnak for $L$-functions of ad\'elic Hilbert modular forms and their convolutions. In particular, under the generalised Riemann hypothesis, we establish several instances…

In math.NT/9907019 we proposed an analog of the classical Riemann hypothesis for characteristic p valued L-series based on the work of Wan, Diaz-Vargas, Thakur, Poonen, and Sheats for the zeta function $\zeta_{\Fr[\theta]}(s)$. During the…

We prove an asymptotic formula for the eighth moment of Dirichlet $L$-functions averaged over primitive characters $\chi$ modulo $q$, over all moduli $q\leq Q$ and with a short average on the critical line. Previously the same result was…

Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good…

We study the 2k-th power moment of Dirichlet L-functions L(s,\chi) at the centre of the critical strip (s=1/2), where the average is over all primitive characters \chi (mod q). We extend to this case the hybrid Euler-Hadamard product…

Friedberg, Hoffstein and Lieman have constructed two related multiple Dirichlet series from quadratic and higher-order $L$-functions and Gauss sums. We compute these multiple Dirichlet series explicitly in the case of the rational function…

Assuming the Generalized Riemann Hypothesis, we provide uniform upper and lower bounds with explicit main terms for $\log{\left|\cL(s)\right|}$ for $\sigma \in (1/2,1)$ and for functions in the Selberg class. In particular, we focus on the…

We establish an analogue of a conjecture of Balasubramanian, Conrey, and Heath-Brown for the family of all Dirichlet characters with conductor up to $Q$. This forms another application of our work in developing an asymptotic large sieve.

We explicitly construct a diffeomorphic pair (p(x),p^{-1}(x)) in terms of an appropriate quadric spline interpolating the prime series. These continuously differentiable functions are the smooth analogs of the prime series and the prime…