Vivek V. Rane
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.
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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.
Using elementary methods,we obtain simple,explicit expressions and bounds of higher order derivatives of Hurwitz zeta function and consequently those of Dirichlet L-function and also,of Lerch's Zeta function at unity (and at Zero too)and also obtain their interrelations.We also state elementary complete forms of approximate functional equations of Hurwitz zeta function and Dirichlet L-function in the critical strip.
We compute the fourth moment of Dirichlet L-functions at the central point for prime moduli, with a power savings in the error term.
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
Twisted fourth moment of Dirichlet $L$-functions to a fixed modulus
本文研究固定素数幂模数下 Dirichlet $L$-函数族在临界线上的扭曲四阶矩，并给出带幂次节省误差项的渐近公式。
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.
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 fourth moment of quadratic Dirichlet $L$-Functions II
本研究无条件证明二次Dirichlet $L$-函数四次矩的含四个主项的渐近公式，基于已有工作引入改进大筛估计、交叉抵消分析等新要素，采用精细组合论证完成证明。
The mollified fourth moment of Dirichlet $L$-functions
该研究推导了带幂节省误差项的Dirichlet $L$-函数族软化四阶矩渐近公式，适用更广模集合且精度更高，改进了中心点四阶矩的已有渐近估计结果。
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…
We study quadratic forms that can occur as trace forms of Galois field extensions L/K, under the assumption that K contains a primitive 4th root of unity. M. Epkenhans conjectured that any such form is a scaled Pfister form. We prove this conjecture and classify the finite groups G which admit a G-Galois extension L/K with a non-hyperbolic trace form. We also give several applications of these results.
Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications
本文建立了Dirichlet L函数的新互反公式，将单个Dirichlet L函数四阶矩与Hecke-Maass新形式中心L值三阶矩关联，推广了Iwaniec、Jutila和Heath-Brown的结果，是Petrow-Young工作的逆版本。
This paper improves the upper bound for the exceptional zeroes of Dirichlet L-functions with even characters. The result is obtained by improving on explicit estimate for $L'(σ;χ)$ for $σ$ close to unity, using a result on the average of Dirichlet characters, and on the lower bound for $L(1;χ)$, with computational aid.
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.
Amplified Fourth Moment of the Riemann Zeta-Function and Applications
本文针对黎曼ζ函数扭曲四阶矩研究的现有局限，推导得到ζ函数四阶矩乘以放大子四次方的渐近公式，可支撑ζ函数零点间隙、矩下界等相关理论研究。
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…
Recently S. Laporta published a partial result on the fourth order QED contribution to the electron anomalous magnetic moment $g-2$. This result contains explicit polylogarithmic parts with fourth and sixth roots of unity. In this note we convert Laporta's result into the motivic `$f$ alphabet'. This provides a much shorter expression which makes the Galois structure visible. We conjecture the $Q$ vector spaces of Galois conjugates of the QED $g-2$ up to weight four. The conversion into the $f$ alphabet relies on a conjecture by D. Broadhurst that iterated integrals in certain Lyndon words pro…
Simultaneous nonvanishing of Dirichlet $L$-functions in Galois orbits
在广义黎曼假设下，证明固定伽罗瓦轨道中正比例特征使两个 $L$-函数乘积在 $1/2$ 处同时非零，并给出磨光矩估计及无条件二次矩计算。
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…
Let χ be the primitive Dirichlet character of conductor 49 defined by χ(3)=ζ, for ζ a primitive 42nd root of unity. We explicitly compute the slopes of the U_7 operator acting on the space of overconvergent modular forms on X_1(49) with weight k and character either χ^{7k-6} or χ^{8-7k}, depending on the embedding of Q(ζ) into C_7. By applying results of Coleman, and of Cohen-Oesterlé, we are then able to conclude the slopes of U_7 acting on all classical Hecke newforms of the same weight and character.
On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters
该研究针对r≥3的Hecke L-函数，利用多重Dirichlet级数理论，推导其在含2r次单位根的整体域上的扭曲矩渐近公式，证明其中心值存在正比例非零值。
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…
In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers $\mathbb N$ as limiting values of $q$-series as $q\to ζ$ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of $\mathbb N$ by analogous structures in the integer partitions $\mathcal P$. In recent work, Wang obtains a wide generalization of Alladi's original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of…
Prime Solutions to a Binary Additive Equation and Mixed Moments of Character Sums
本文通过泊松求和与Kloosterman分数和均值估计，得到二元加法方程素数解的渐近公式，并应用于特征和混合四阶矩，获得非平凡界。
Non-vanishing for cubic Hecke $L$-functions
本文研究了三次海克勒L-函数在中心点处非消失性的无条件结果，通过 mollified 短时域方法，首次给出了三次族的渐近评估并建立了新的上界。
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