Sandro Bettin, J. Brian Conrey
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.
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We show a general lower bound for Mean-value of Dirichlet polynomials
In the present paper, we introduce Eulerian polynomials attached to by using p-adic q-integral on Zp . Also, we give new interesting identities via the generating functions of Dirichlet's type of Eulerian polynomials. After, by applying Mellin transformation to this generating function of Dirichlet' type of Eulerian polynomials, we derive L-function for Eulerian polynomials which interpolates of Dirichlet's type of Eulerian polynomials at negative integers.
Conditionally on the Riemann hypothesis we prove asymptotic formulae for mean values of various long Dirichlet polynomials involving the von Mangoldt function. Our results avoid the use of correlation sum estimates although in addition to the Riemann hypothesis we must assume that our Dirichlet polynomials have weights from a specific class whose transforms are sufficiently concentrated near the origin. We also give large deviation estimates for these long Dirichlet polynomials.
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 prove that suitable properties of the twists by Dirichlet characters of an L-function of degree 2 imply that its Euler product is of polynomial type.
A note on Dirichlet-like series attached to polynomials
该研究引入并研究了附属于周期函数与多项式对的类Dirichlet函数，其推广了标准Dirichlet L-函数，具备全复平面全纯、负整数处取值简单的相似性质。
In this paper we obtain some new estimates for the number of large values of Dirichlet polynomials. Our results imply new zero density estimates for the Riemann zeta function which give a small improvement on results of Bourgain and Jutila.
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…
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.
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.
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.
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.
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 $\log\log\log{x}/\sqrt{\log\log{x}}$.
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。
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。
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。
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…
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.
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 …
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。
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。
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 $(\log N)^{1/4}(\log\log N)^{1/4}$.
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.
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。
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。
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。
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.
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…
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.
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 $J_s$.
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 $p\nmid q_F$
In this study, we construct the two-variable multiple Dirichlet q-L-function and two-variable multiple Dirichlet type Changhee q-L-function. These functions interpolate the q-Bernoulli polynomials and generalized Changhee q-Bernoulli polynomials. By using the Mellin transformation, we give an integral representation for the two-variable multiple Dirichlet type q-zeta function and the two variable multiple Dirichlet type Changhee q-L-function.
Let $π$ be a cuspidal representation on $\GL(2,\mathbb{A}_{\mathbb{Q}}).$ We give nontrivial lower and upper bounds for average of absolute values of Dirichlet coefficients associated to $π;$ and nontrivial upper bound in the case of $\Sym^kπ,$ $k=2, 3.$ These bounds generalize the known estimates in holomorphic case to Maass forms, without assuming Ramanujan-Petersson conjecture.
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.
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…
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。
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。
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。
Let D be a square-free polynomial in F_q[t], where q is odd, and let G be a genus of definite ternary lattices over F_q[t] of determinant D. In this paper we give self-contained and relatively elementary proofs of Siegel's formulas for the weighted sum of primitive representations numbers over the classes of G and for the mass of G. Our proof of the mass formula shows an interesting relation with certain averages of Dirichlet L-functions.
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…
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.
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.
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。
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
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…
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, $$ \exp( (\log N)^{1/2-\varepsilon} ) \ll \sup_{|t| \leq N^C} |D_N(t)| \ll \exp( (\log N)^{1/2+\varepsilon}). $$
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].
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.
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。
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 $f$.
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。
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.
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。
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.
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。
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.
The eighth moment of Dirichlet L-functions II
本研究无条件证明了模q≤Q的本原特征下、临界线短平均的Dirichlet L函数第八矩渐近公式，突破了此前该结果依赖广义黎曼假设的限制。
The mixed second moment of prime quadratic Dirichlet $L$-functions over function fields
该研究在函数域背景下，针对首一不可约多项式对应的二次Dirichlet L-函数的μ阶和ν阶导数，计算出其混合二阶矩的渐近公式，为相关L-函数导数矩的研究提供了结果。
We introduce Möbius functions of higher rank, a new class of arithmetic functions so that the classical Möbius function is of rank 2. With this idea, we evaluate Dirichlet series on the sum of the reciprocal square of all $r$-free numbers. For the proof, the Riemann zeta function and cyclotomic polynomials play a key role.
The Mixed Second Moment of Quadratic Dirichlet $L$-functions with Prime Conductors
本文在函数域语境下，针对首一不可约多项式平均的情况，建立了含完备L函数二阶导数的混合二阶矩的渐近公式。
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