We prove a Lindelöf-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 $q^2|(q^*)^3$. As a consequence, we finish the previous work of the authors and establish a Weyl-strength subconvex bound for all Dirichlet $L$-functions with no restrictions on the conductor.
Extending a result of Heath-Brown, we establish an asymptotic formula for the fourth moment of central values of Dirichlet $L$-functions attached to primitive characters $\pmod q$.
We study the moments of the Dirichlet L-function when defined over the polynomial ring over finite fields. We find an asymptotic formula to the fourth moment of the central value of Dirichlet L functions in this context. We also find a lower bound to the 2kth moment of these L-functions.
We obtain by simple elementary method, the character average of fourth power of Dirichlet L-series at unity.This result seems to be new.We also obtain character average of second power of Dirichlet L-series at unity,using the power series in second variable of Hurwitz zeta function at unity.
For a positive integer $q\not\equiv 2 \pmod 4$, this work considers the fourth moment of Dirichlet $L$-functions averaged over both $t\in [0,T]$ and primitive characters to modulus $q$. An asymptotic formula with a power saving from both $q$-aspect and $t$-aspect in the error term is obtained.
We prove an asymptotic formula for the fourth power mean of Dirichlet L-functions averaged over primitive characters to modulus q and over t\in [0,T] which is particularly effective when q \ge T. In this range the correct order of magnitude was not previously known.
We study the fourth moment of quadratic Dirichlet $L$-functions at $s= \frac{1}{2}$. We show an asymptotic formula under the generalized Riemann hypothesis, and obtain a precise lower bound unconditionally. The proofs of these results follow closely arguments of Soundararajan and Young [19] and Soundararajan [17].
We prove an asymptotic formula for the second moment of a product of two Dirichlet L-functions on the critical line, which has a power saving in the error term and which is uniform with respect to the involved Dirichlet characters. As special cases we give uniform asymptotic formulae for the fourth moment of individual Dirichlet L-functions and for the second moment of Dedekind zeta functions of quadratic number fields on the critical line.
We evaluate some twisted fourth moment of Dirichlet $L$-functions at the central point s=1/2 and for prime moduli q. The principal tool is a careful analysis of a shifted convolution problem involving the divisor function using spectral theory of automorphic forms and bounds for bilinear forms in Kloosterman sums. Having in mind simultaneous non vanishing results, we apply the Theorem to establish an asymptotic formula of a mollified fourth moment for this family of $L$-functions
We obtain an asymptotic formula for the fourth moment of quadratic Dirichlet $L$--functions over $\mathbb{F}_q[x]$, as the base field $\mathbb{F}_q$ is fixed and the genus of the family goes to infinity. According to conjectures of Andrade and Keating, we expect the fourth moment to be asymptotic to $q^{2g+1} P(2g+1)$ up to an error of size $o(q^{2g+1})$, where $P$ is a polynomial of degree $10$ with explicit coefficients. We prove an asymptotic formula with the leading three terms, which agrees with the conjectured result.
The asymptotic formula of the fourth moment of Dirichlet $L$-functions at the central value was predicted in a conjecture by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, and the prime moduli case was proved by M. P. Young in 2011. This work establishes this asymptotic formula for general moduli. The work relies on the study of a special divisor sum function, called $\mathcal{D}_q$-function, which plays a key role in deducing the main terms. Another key ingredient is an application of new bounds for double sums in Kloosterman sums, applied in the estimate of th…
We prove results on moments of $L$-functions in the function field setting, where the moment averages are taken over primitive characters of modulus $R$, where $R$ is a polynomial in $\mathbb{F}_q [T]$. We consider the behaviour as $\textrm{deg} R \rightarrow \infty$ and the cardinality of the finite field is fixed. Specifically, we obtain an exact formula for the second moment provided that $R$ is square-full, and an asymptotic formula for the fourth moment for any $R$. The fourth moment result is a function field analogue of Heath-Brown's result in the number field setting, which was subsequ…
We obtain the asymptotic main term of moments of arbitrary derivatives of $L$-functions in the function field setting. Specifically, the first, second, and mixed fourth moments. The average is taken over all non-trivial characters of a prime modulus $Q \in \mathbb{F}_q [t]$, and the asymptotic limit is as $\mathrm{deg} \, Q \longrightarrow \infty$. This extends the work of Tamam who obtained the asymptotic main term of low moments of $L$-functions, without derivatives, in the function field setting. It also expands on the work of Conrey, Rubinstein, and Snaith who cojectured, using random matr…
Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic L-functions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet L-functions modulo q weighted by a non-archimedean test function. This establishes a new reciprocity formula. As an application, we obtain sharp upper bounds for the fourth moment twisted by the square of a Dirichlet polynomial of length q^{1/4}. An auxiliary result of independent interest is a sharp upper …
We prove an asymptotic for the eighth moment of Dirichlet $L$-functions averaged over primitive characters $χ$ modulo $q$, over all moduli $q\leq Q$ and with a short average on the critical line, conditionally on GRH. We derive the analogous result for the fourth moment of Dirichlet twists of GL(2) L-functions. Our results match the moment conjectures in the literature; in particular, the constant 24024 appears as a factor in the leading order term of the eighth moment.
We extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized cubic moment of modular L-functions at the central point s = 1/2 and for prime moduli q. As an application, we exploit our recent result on the mollification of the fourth moment of Dirichlet L-functions to derive that for any pair $(ω_1,ω_2)$ of multiplicative characters modulo q, there is a positive proportion of $χ$ (mod q) such that $L(χ, 1/2 ), L(χω_1, 1/2 )$ and $L(χω_2, 1/2)$ are simultaneo…
We examine the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. Previously this approach has proved unsuccessful in computing moments beyond the eighth, even heuristically. A careful analysis of the second and fourth moments illustrates the nature of the problem and enables us to identify the terms that are missed in the standard application of these methods.
This is part II of our examination of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations.
We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length $T^{\frac14 - \varepsilon}$. Our work relies crucially on Watt's theorem on averages of Kloosterman fractions. In the context of the twisted fourth moment, Watt's result is an optimal replacement for Selberg's eigenvalue conjecture. Our work extends the previous result of Hughes and Young, where Dirichlet polynomials of length $T^{\frac{1}{11}-\varepsilon}$ were considered. Our result has several applications, among others to the proportion of critical zeros of the Riemann zeta…
Let $W$ be the Weyl group of a simply-laced affine Kac-Moody Lie group, excepting $\tilde{A}_n$ for $n$ even. We construct a multiple Dirichlet series $Z(x_1, \ldots x_{n+1})$, meromorphic in a half-space, satisfying a group $W$ of functional equations. This series is analogous to the multiple Dirichlet series for classical Weyl groups constructed by Brubaker-Bump-Friedberg, Chinta-Gunnells, and others. It is completely characterized by four natural axioms concerning its coefficients, axioms which come from the geometry of parameter spaces of hyperelliptic curves. The series constructed this w…
Let $χ$ range over the $(p-1)/2$ even Dirichlet characters modulo a prime $p$ and denote by $θ(x,χ)$ the associated theta series. The asymptotic behaviour of the second and fourth moments proved by Louboutin and the author implies that there exists at least $ \gg p/ \log p$ characters such that the associated theta function does not vanish at a fixed point. Constructing a suitable mollifier, we improve this result and show that there exists at least $ \gg p/ \sqrt{\log p}$ characters such that $θ(x,χ) \neq 0$ for any $x>0$. We give similar results for odd Dirichlet characters mod $p$.
We prove a theorem that allows one to count solutions to determinant equations twisted by a periodic weight with high uniformity in the modulus. It is obtained by using spectral methods of $\operatorname{SL}_2(\mathbb{R})$ automorphic forms to study Poincaré series over congruence subgroups. By keeping track of interactions between multiple orbits we get advantages over the widely used sums of Kloosterman sums techniques. We showcase this with applications to correlations of the divisor functions twisted by periodic functions and the fourth moment of Dirichlet $L$-functions on the critical lin…
We study the average of the product of the central values of two $L$-functions of modular forms $f$ and $g$ twisted by Dirichlet characters to a large prime modulus $q$. As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of $q$, and we use the theory of Deligne and Katz to estimate certain complete exponential sums in several variables and prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of $q$. When at least one of the forms $f$ an…
This is a rework of our old file, which has been left unpublished since September 1994, on an explicit spectral decomposition of the fourth power moment of the Riemann zeta-function against a weight which is the square of a Dirichlet polynomial. At this occasion we add an explicit treatment of generalized Kloosterman sums associated with arbitrary Hecke congruence subgroups (Section 15), which might have an independent interest. At the end (Section 36) of our discussion, we set out a few problems on the distribution of eigenvalues of the hyperbolic Laplacian, which appear to us to be related t…
Let $Δ(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|ζ(1/2 + it)|$. If $E^*(t) = E(t) - 2πΔ^*(t/(2π))$ with $Δ^*(x) = - Δ(x) +2Δ(2x)- {1\over2}Δ(4x)$, then we obtain $$ \int_0^T(E^*(t))^4 dt \ll_εT^{16/19+varepsilon}$$, which is the first non-trivial bound for higher moments of $E^*(t)$. The method of proof also provides an upper bound for sums of fourth powers of mean square integrals of $|ζ(1/2 + it)|$ over well-spaced points. This, in turn, yields a new proof of the twelfth moment estimate for $|ζ(1/2…
