On Mean Values of Dirichlet Polynomials

Zeros of Dirichlet polynomials

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$ and study the distribution of zeros of Dirichlet polynomials $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ corresponding to these functions. We prove that the known non-trivial zero-free half plane for Dirichlet polynomials associated to this class of multiplicative functions is optimal. We also introduce a characterization of elements in this class based on a new parameter depending on the Dirichlet series $F(s) = \sum_{n=1}^\infty f(n) n^{-s}$. In this context, we obtain non-trivial regions in which the associa…

Mean values of long Dirichlet polynomials with higher divisor coefficients

In this article, we prove an asymptotic formula for mean values of long Dirichlet polynomials with higher order shifted divisor functions, assuming a smoothed additive divisor conjecture for higher order shifted divisor functions. As a consequence of this work, we prove special cases of conjectures of Conrey-Keating on mean values of long Dirichlet polynomials with higher order shifted divisor functions as coefficients.

Dirichlet's type of twisted Eulerian polynomials in connection with Eulerian-L-function

In the present paper, we effect Dirichlet's type of twisted Eulerian polynomials by using p-adic fermionic q-integral on the p-adic integer ring. Also, we introduce some new interesting identities for them. As a result of them, by using contour integral on the generating function of Dirichlet's type of twisted Eulerian polynomials and so we define twisted Eulerian-L-function which interpolates of Dirichlet's type of Eulerian polynomials at negative integers which we state in this paper.

Zeros of Dirichlet Polynomials via a Density Criterion

We obtain a necessary and sufficient condition in order that a semi-plane of the form $\Re(s)>r$, $r\in \mathbb{R}$, is free of zeros of a given Dirichlet polynomial. The result may be considered a natural generalization of a well-known criterion for the truth of the Riemann hypothesis due to Báez-Duarte. An analog for the case of Dirichlet polynomials of a result of Burnol which is closely related to Báez-Duarte's one is also established.

Dirichlet's Theorem for polynomial rings

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many polynomials c(X)\in F[X] such that a(X) + b(X)c(X) is irreducible of degree n, provided that F has a separable extension of degree n.

On the Sidon Constant for Dirichlet Polynomials

We estimate the error term in the asymptotic formula of the Sidon constant for (ordinary) Dirichlet polynomials by providing explicit lower and upper bounds. The lower bound is already implicitly known, but we supply the necessary computations to make it explicit. The upper bound is improved by a factor of $\log\log\log{x}/\sqrt{\log\log{x}}$.

通过短Dirichlet多项式近似SL(3,Z) Hecke-Maass L函数

Approximations of $SL(3,\mathbb{Z})$ Hecke-Maass $L$-Functions by short Dirichlet polynomials

本文通过短Dirichlet多项式近似SL(3,Z) Hecke-Maass L函数，利用Kuznetsov迹公式方法，展示了L函数倒数的平均近似性。

关于零密度结果蕴含 Dirichlet 多项式大值估计的一个注记

A note on zero density results implying large value estimates for Dirichlet polynomials

本文证明黎曼 zeta 函数的零密度估计可蕴含 Dirichlet 多项式的大值估计，填补了反向蕴含关系的空白。

带除数系数的长Dirichlet多项式的均值

Mean values of long Dirichlet polynomials with divisor coefficients

本文研究带除数系数的长光滑Dirichlet多项式的均值问题，借助已有一般性定理推导出含所有低阶项与幂节省误差项的渐近公式，在除数函数情形下强化了Conrey与Gonek的猜想。

Local Suprema of Dirichlet Polynomials and Zerofree Regions of the Riemann Zeta-Function

A new zerofree region of the Riemann Zeta-function $ζ$ is identified by using Turán's localization criterion linking zeros of $ζ$ with uniform local suprema of sets of Dirichlet polynomials expanded over the primes. The proof is based on a randomization argument. An estimate for local extrema for some finite families of shifted Dirichlet polynomials, % associated to linearly independent sequences of reals is established by preliminary considering their local increment properties, by means of Montgomery-Vaughan's variant of Hilbert's inequality. A covering argument combined with Turán's localiz…

Infinite products involving Dirichlet characters and cyclotomic polynomials

Using some basic properties of the gamma function, we evaluate a simple class of infinite products involving Dirichlet characters as a finite product of gamma functions and, in the case of odd characters, as a finite product of sines. As a consequence we obtain evaluations of certain multiple $L$-series. In the final part of this paper we derive expressions for infinite products of cyclotomic polynomials, again as finite products of gamma or of sine functions.

The mean square of the product of $ζ(s)$ with Dirichlet polynomials

Improving earlier work of Balasubramanian, Conrey and Heath-Brown, we obtain an asymptotic formula for the mean-square of the Riemann zeta-function times an arbitrary Dirichlet polynomial of length $T^{1/2 + δ}$, with $δ= 0.01515....$ As an application we obtain an upper bound of the correct order of magnitude for the third moment of the Riemann zeta-function. We also refine previous work of Deshouillers and Iwaniec, obtaining asymptotic estimates in place of bounds. Using the work of Watt, we compute the mean-square of the Riemann zeta-function times a Dirichlet polynomial of length going up …

Dirichlet多项式的大值估计新结果

New large value estimates for Dirichlet polynomials

本文通过改进Dirichlet多项式大值估计，得出了长度为N的多项式取值接近N^{3/4}的更优界，从而推导出零密度估计和短区间内素数渐进行为。

Averages of long Dirichlet polynomials

We consider the asymptotic behavior of the mean square of truncations of the Dirichlet series of $ζ(s)^k$. We discuss the connections of this problem with that of the variance of the divisor function in short intervals and in arithmetic progressions, reviewing the recent results on this topic. Finally, we show how these results can all be proved assuming a suitable version of the moments conjecture.

关于除数函数多项式与狄利克雷级数

On Divisor Function Polynomials and Dirichlet Series

该研究针对和集系计数函数推导狄利克雷级数与收敛界，定义并研究了除数函数多项式，揭示其与伯努利数、有序贝尔多项式的关联。

Upper bounds for $L^q$ norms of Dirichlet polynomials with small $q$

We improve on previous upper bounds for the $q$th norm of the partial sums of the Riemann zeta function on the half line when $0<q\leqslant 1$. In particular, we show that the 1-norm is bounded above by $(\log N)^{1/4}(\log\log N)^{1/4}$.

Mean values of Dirichlet polynomials and applications to linear equations with prime variables

We prove a new mean-value theorem for Dirichlet polynomials with coefficients given by the von Mangoldt function. We then use our theorem to derive new estimates for certain exponential sums over primes. The latter have applications to additive problems with prime variables. In particular, we are able to improve on a recent result of J.Y. Liu and K.M. Tsang on the size of the solutions of linear equations with prime variables.

临界带中黎曼zeta函数与Dirichlet多项式乘积的均方值

The mean square of the product of the Riemann zeta-function and a Dirichlet polynomial in the critical strip

本文改进了Matsumoto和Ishikawa的工作，通过获得Atkinson型误差项显式公式，给出了临界带内黎曼zeta函数与Dirichlet多项式乘积均方值的渐近公式，与广义Dirichlet除数问题紧密相关。

带模系数的长 Dirichlet 多项式的平均值

Averages of long Dirichlet polynomials with modular coefficients

本文在广义 Lindelöf 假设下，研究了原始尖点形式相关 L 函数在权重方面的矩，并得到了带模系数的长 Dirichlet 多项式扭曲矩的渐近公式，该结果与 CFKRS 配方的预测一致。

长Dirichlet多项式的二次扭曲的平均

Averages of quadratic twists of long Dirichlet polynomials

本文在Lindelöf假设下计算了由Kronecker符号扭曲的长Dirichlet多项式的平均值，验证了其与[CFKRS]配方中的0-和1-交换项一致。

p-adic interpolating function associated with modified Dirichlet's type of twisted q-euler numbers and polynomials with weight alpha

The q-calculus theory is a novel theory that is based on finite difference re-scaling. The rapid development of q-calculus has led to the discovery of new generalizations of q-Euler polynomials involving q-integers. The present paper deals with the modified Dirichlet's type of twisted q-Euler polynomials with weight alpha. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of q-Euler numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between mod…

Breaking the $\frac{1}{2}$-barrier for the twisted second moment of Dirichlet $L$-functions

We study the second moment of Dirichlet $L$-functions to a large prime modulus $q$ twisted by the square of an arbitrary Dirichlet polynomial. We break the $\frac{1}{2}$-barrier in this problem, and obtain an asymptotic formula provided that the length of the Dirichlet polynomial is less than $q^{51/101} = q^{1/2 +1/202}$. As an application, we obtain an upper bound of the correct order of magnitude for the third moment of Dirichlet $L$-functions. We give further results when the coefficients of the Dirichlet polynomial are more specialized.

Twists by Dirichlet characters and polynomial Euler products of L-functions, II

In a previous paper we proved that if an $L$-function $F$ from the Selberg class has degree $2$, its conductor $q_F$ is a prime number and $F$ is weakly twist-regular at all primes $p\neq q_F$, then $F$ has a polynomial Euler product. In this paper we extend this result to $L$-functions of degree 2 with square-free conductor $q_F$, which are weakly twist-regular at all primes $p\nmid q_F$

An optimal choice of Dirichlet polynomials for the Nyman-Beurling criterion

We give a conditional result on the constant in the Báez-Duarte reformulation of the Nyman-Beurling criterion for the Riemann Hypothesis. We show that assuming the Riemann hypothesis and that $\sum_ρ\frac{1}{|ζ'(ρ)|^2}\ll T^{3/2-δ}$, for some $δ>0$, the value of this constant coincides with the lower bound given by Burnol.

The Integral Moments and Ratios of Quadratic Dirichlet $L$-Functions over Monic Irreducible Polynomials in $\mathbb{F}_{q}[T]$

In this paper we extend to the function field setting the heuristics formerly developed by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments of $L$-functions. We also adapt to the function setting the heuristics first developed by Conrey, Farmer and Zirnbauer to the study of mean values of ratios of $L$-functions. Specifically, the focus of this paper is on the family of quadratic Dirichlet $L$-functions $L(s,χ_{P})$ where the character $χ$ is defined by the Legendre symbol for polynomials in $\mathbb{F}_{q}[T]$ with $\mathbb{F}_{q}$ a finite field of odd cardinality and…

具有乘性系数的 Dirichlet 多项式的大值

Large values of Dirichlet polynomials with multiplicative coefficients

本文研究了具有乘性系数的 Dirichlet 多项式的大值问题，在特定区域证明了改进的 Omega 结果，并在 $\log N$ 位于 $\sqrt{\log T\log_2T}$ 附近时给出了 Omega 结果。

关于用Dirichlet多项式逼近zeta函数

On the approximation of the zeta function by Dirichlet polynomials

该研究在数论领域改进了zeta函数的Dirichlet多项式逼近余项估计，将误差常数优化为29/14，提升了Titchmarsh经典定理的结果精度。

Dirichlet多项式的大值估计与短区间上的三线性和

Large value estimates for Dirichlet polynomials and trilinear sums over short intervals

本文借鉴Matomäki等人的方法，对$\ rac{1}{2} < \ heta < \ rac{4}{7}$改进了短区间三线性和的Baker--Harman--Pintz估计，得到$\ rac{1}{2} < \ heta < 1$的新结果，可改进Li关于短区间素数的研究。

Algebra of formal power series, isomorphic to the algebra of formal Dirichlet series

Ordinary algebra of formal power series in one variable is convenient to study by means of the algebra of Riordan matrices and the Riordan group. In this paper we consider algebra of formal power series without constant term, isomorphic to the algebra of formal Dirichlet series. To study it, we introduce matrices, similar to the Riordan matrices. As a result, some analogies between two algebras becomes visible. For example, the Bell polynomials (polynomials of partitions of number $n$ into $m$ parts) play a certain role in the ordinary algebra. Similar polynomials (polynomials of decomposition…

Supremum of Random Dirichlet Polynomials with Sub-multiplicative Coefficients

We study the supremum of random Dirichlet polynomials $D_N(t)=\sum_{n=1}^N\varepsilon_n d(n) n^{- s}$, where $(\varepsilon_n)$ is a sequence of independent Rademacher random variables, and $ d $ is a sub-multiplicative function. The approach is gaussian and entirely based on comparison properties of Gaussian processes, with no use of the metric entropy method.

Character sums over products of prime polynomials

We study sums of Dirichlet characters over polynomials in $\mathbb{F}_q[t]$ with a prescribed number of irreducible factors. Our main results are explicit formulae for these sums in terms of zeros of Dirichlet L-functions. We also exhibit new phenomena concerning Chebyshev-type biases of such sums when the number of irreducible factors is very large.

q-Ehrhart polynomials of Gorenstein polytopes, Bernoulli umbra and related Dirichlet series

This article considers some q-analogues of classical results concerning the Ehrhart polynomials of Gorenstein polytopes, namely properties of their q-Ehrhart polynomial with respect to a good linear form. Another theme is a specific linear form Ψ (involving Carlitz' q-analogues of Bernoulli numbers) on the space of polynomials, for which one shows interesting behaviour on these q-Ehrhart polynomials. A third point is devoted to some related zeta-like functions associated with polynomials

主 Dirichlet 特征标的 Fekete 多项式

Fekete polynomials of principal Dirichlet characters

本文研究主 Dirichlet 特征标的 Fekete 多项式，刻画其分圆因子并分析非分圆部分的不可约性、可分性、系数、特殊值及 Galois 群。

On the families of q-Euler numbers and polynomials and their applications

In the present paper, we investigate special generalized q-Euler numbers and polynomials. Some earlier results of T. Kim in terms of q-Euler polynomials with weight alpha can be deduced. For presentation of our formulas we apply the method of generating function and p-adic q-integral representation on Zp. We summarize our results as follows. In section 2, by using combinatorial techniques we present two formulas for q-Euler numbers with weight alpha. In section 3, we derive distribution formula (Multiplication Theorem) for Dirichlet type of q-Euler numbers and polynomials with weight . Moreove…

Extremal bounds for Dirichlet polynomials with random multiplicative coefficients

For $X(n)$ a Steinhaus random multiplicative function, we study the maximal size of the random Dirichlet polynomial $$ D_N(t) = \frac1{\sqrt{N}} \sum_{n \leq N} X(n) n^{it}, $$ with $t$ in various ranges. In particular, for fixed $C>0$ and any small $\varepsilon>0$ we show that, with high probability, $$ \exp( (\log N)^{1/2-\varepsilon} ) \ll \sup_{|t| \leq N^C} |D_N(t)| \ll \exp( (\log N)^{1/2+\varepsilon}). $$

Lagrange Inversion Theorem for Dirichlet series

We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4].

Dirichlet polynomials: some old and recent results, and their interplay in number theory

In the first part of this expository paper, we present and discuss the interplay of Dirichlet polynomials in some classical problems of number theory, notably the Lindelöf Hypothesis. We review some typical properties of their means and continue with some investigations concerning their supremum properties. Their random counterpart is considered in the last part of the paper, where a analysis of their supremum properties, based on methods of stochastic processes, is developed.

Dirichlet多项式的零密度集中

Zero-Density Concentration for Dirichlet Polynomials

本文证明有限Dirichlet和的零点垂直密度渐近集中于单条直线，给出一般准则并应用于zeta函数部分和、Dirichlet L-函数及Hecke本征形式，集中线为Re s=1/2或k/2。

Extreme values of Dirichlet polynomials with multiplicative coefficients

We study extreme values of Dirichlet polynomials with multiplicative coefficients, namely \[D_N(t) : = D_{f,\, N}(t)= \frac{1}{\sqrt{N}} \sum_{n\leqslant N} f(n) n^{it}, \] where $f$ is a completely multiplicative function with $|f(n)|=1$ for all $n\in\mathbb{N}$. We use Soundararajan's resonance method to produce large values of $\left|D_N(t)\right|$ uniformly for all such $f$. In particular, we improve a recent result of Benatar and Nishry, where they establish weaker lower bounds and only for almost all such $f$.

Dirichlet $L$-函数的软化四阶矩

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 Fourth Moment of Dirichlet L-Functions for the Rational Function Field

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.

二次狄利克雷 $L$-函数在 $\mathbb{F}_q[t]$ 上首一不可约多项式处的大值

Large values of quadratic Dirichlet $L$-functions over monic irreducible polynomial in $\mathbb{F}_q[t]$

本文证明了大亏格极限下二次狄利克雷 $L$-函数在首一不可约多项式处的 $\Omega$-结果，其下界与 Bondarenko--Seip 界同阶。

Cauchy Means of Dirichlet polynomials

We study Cauchy means of Dirichlet polynomials $$\int_\R \Big|\sum_{n=1}^N \frac{1}{ n^{\s+ ist}} \Big|^{2q} \frac{\dd t}{π( t^2+1)}.$$ These integrals were investigated when $q=1,\s= 1, s=1/2 $ by Wilf, using integral operator theory and Widom's eigenvalue estimates. We show the optimality of some upper bounds obtained by Wilf. We also obtain new estimates for the case $q\ge 1$, $\s\ge 0$ and $s>0$. We complete Wilf's approach by relating it with other approaches (having notably connection with Brownian motion), allowing simple proofs, and also prove new results.

具有特征的狄利克雷多项式的大值估计和狄利克雷\(L\)函数的零点密度

Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet $L$-Functions

研究狄利克雷多项式大值估计及狄利克雷\(L\)函数零点密度，通过改编Guth--Maynard方法改进指数，得到零点密度估计，进而推导模素数算术级数中最小哥德巴赫数新上界及短区间内算术级数中素数的新结果。

关于Dirichlet $L$-函数在正整数处乘积的均值

On the mean values of products of Dirichlet $L$-functions at positive integers

本文研究Dirichlet $L$-函数在正整数处乘积的均值分布，给出了由Dirichlet特征加权、涉及Bernoulli函数和Jordan totient函数的显式公式，推广了多种已知结果。

The mean-square of Dirichlet L-functions

We verify the conjecture of [CFKRS] for the mean square near the critical point of Dirichlet L-functions for a composite modulus q. We also prove a kind of reciprocity formula when the second moment for a prime modulus is twisted by a character evaluated at a different prime.

Mean values of cubic and quartic Dirichlet Characters

We evaluate the average of cubic and quartic Dirichlet character sums with the modulus going up to a size comparable to the length of the individual sums. This generalizes a result of Conrey, Farmer and Soundararajan on quadratic Dirichlet character sums.

On two mean square averages of Dirichlet $L$-function

Finding the mean square averages of the Dirichlet $L$-functions over Dirichlet characters $χ$ of same parity is an active problem in number theory. Here we explicitly evaluate such averages of $L(3,χ)$ and $L(4,χ)$ using certain trigonometric sums and Bernoulli polynomials and express them in terms of the Euler totient function $ϕ$ and the Jordan totient function $J_s$.

A generalization of Tóth identity in $\mathbb{F}_q[T]$ involving a Dirichlet Character

Let $\mathbb{A}=\mathbb{F}_q[T]$ be the polynomial ring over the finite field $\mathbb{F}_q$. In this article, we prove a generalization of Tóth identity on $\mathbb{A}$ involving arithmetical functions, multiplicative and additive characters.

Dirichlet polynomials, Majorization, and Trumping

Majorization and trumping are two partial orders which have proved useful in quantum information theory. We show some relations between these two partial orders and generalized Dirichlet polynomials, Mellin transforms, and completely monotone functions. These relations are used to prove a succinct generalization of Turgut's characterization of trumping.

整数、多项式和排列分解的狄利克雷定律

Dirichlet law for factorization of integers, polynomials and permutations

本文通过多维围道积分证明整数、多项式和排列分解为 $k$ 部分时遵循狄利克雷分布，推广了 DDT 反正弦定律，并指出任意参数的狄利克雷分布可类似建模。

The twisted mean square and critical zeros of Dirichlet $L$-functions

In this work, we obtain an asymptotic formula for the twisted mean square of a Dirichlet $L$-function with a longer mollifier, whose coefficients are also more general than before. As an application we obtain that, for every Dirichlet $L$-function, more than 41.72\% of zeros are on the critical line and more than 40.74\% of zeros are simple and on the critical line. These proportions also improve previous results which were proved only for the Riemann zeta-function.

对数牛顿多边形与多胞体，以及狄利克雷多项式的因式分解

Logarithmic Newton polygons and polytopes, and the factorization of Dirichlet polynomials

本文通过引入对数牛顿多边形与对数牛顿多胞体等工具，绕过典范映射对整数素因子分解的依赖，建立了狄利克雷多项式的一系列不可约性、无平方因子性及绝对不可约性判据，推广了经典多项式因式分解理论。

二元多项式的一个精确族与Chinburg猜想变体

An exact family of bivariate polynomials and Variants of Chinburg's Conjectures

本文通过研究二元多项式族$P_d(x,y)$的Mahler测度，将其表示为狄利克雷$L$-函数的线性组合，从而为Chinburg猜想及其推广版本提供了解答，覆盖多个导子情况。