Hirotaka Kobayashi
We consider a sum of the derivatives of Dirichlet $L$-functions over the zeros of Dirichlet $L$-functions. We give an asymptotic formula for the sum.
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Joint extreme values of Dirichlet (L)-functions and their logarithmic derivatives
本文利用共振方法研究狄利克雷(L)-函数及其对数导数的联合极值，扩展了Aistleitner等人(2019)和Yang(2023)的研究成果。
Large values of logarithmic derivatives of quadratic Dirichlet $L$-functions
本文利用共振方法，推导了二次狄利克雷L函数对数导数的条件Omega结果，改进并扩展了Mortada和Murty以及Yang的先前结果。
Let C(q,+) be the set of even, primitive Dirichlet characters (mod q). Using the mollifier method we show that L^{(k)}(1/2,chi) is not equal to zero for almost all the characters chi in C(q,+) when k and q are large. Here, L^{(k)}(s,chi) is the k-th derivative of of the Dirichlet L-function L(s,chi).
Extreme values of derivatives of Dirichlet $L$-functions
研究狄利克雷L函数导数在1/2<σ<1范围内的极值，通过与前人工作对比，得出该范围内导数能达到与原L函数同量级极值的结论。
Zeros of the Riemann zeta function and its derivatives have been studied by many mathematicians. Among, the number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, Akatsuka, and the author. Berndt, Levinson, and Montgomery investigated the unconditional case, while Akatsuka and the author gave sharper estimates under the truth of the Riemann hypothesis. In this paper, we prove similar results related to the first derivative of the Dirichlet $L$-functions associated with primi…
We compute the adjoint of the Serre derivative map with respect to the Petersson scalar product by using existing tools of nearly holomorphic modular forms. The Fourier coefficients of a cusp form of integer weight $k$, constructed using this method, involve special values of certain shifted Dirichlet series associated with a given cusp form $f$ of weight $k+2$. As application, we get an asymptotic bound for the special values of these shifted Dirichlet series and also relate these special values with the Fourier coefficients of $f$. We also give a formula for the Ramanujan tau function in ter…
Yıldırım has classified zeros of the derivatives of Dirichlet $L$-functions into trivial zeros, nontrivial zeros and vagrant zeros. In this paper we remove the possibility of vagrant zeros for $L'(s,χ)$ when the conductors are large to some extent. Then we improve asymptotic formulas for the number of zeros of $L'(s,χ)$ in $\{s\in\mathbb{C}:\operatorname{Re}(s)>0, |\operatorname{Im}(s)|\leq T\}$. We also establish analogues of Speiser's theorem, which characterize the generalized Riemann hypothesis for $L(s,χ)$ in terms of zeros of $L'(s,χ)$, when the conductor is large.
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…
In this paper, we study the value distribution of the derivative of a Dirichlet $L$-function $L'(s,χ)$ at the $a$-points $ρ_{a,χ}=β_{a,χ}+iγ_{a,χ}$ of $L(s,χ).$ We give an asymptotic formula for the sum $$\sum_{ρ_{a,χ};\ 0<γ_{a,χ}\leq T}L'\left(ρ_{a,χ},χ\right) X^{ρ_{a,χ}}\ \ \hbox{as}\ \ T\longrightarrow \infty,$$ where $X$ is a fixed positive number and $χ$ is a primitive character $\mod q$. This work continues the investigations of Fujii \cite{2,3,4}, Garunk$\check{s}$tis \& Steuding \cite{7} and the authors \cite{12}.
Conditional estimates for the logarithmic derivative of Dirichlet $L$-functions
本文在广义黎曼猜想假设下，对模 q≥10^30 的本原 Dirichlet L-函数的对数导数建立了 q 方面的显式界，并在 σ=1 处改进了 Ihara-Murty-Shimura 的结果，同时给出 Riemann zeta 函数的类似估计。
Under the generalized Lindelöf Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke eigenvalues at Piatetski-Shapiro primes.
Using the Kuznetsov trace formula, we prove a spectral decomposition for the sums of generalized Dirichlet $L$-functions. Among applications are an explicit formula relating norms of prime geodesics to moments of symmetric square $L$-functions and an asymptotic expansion for the average of central values of generalized Dirichlet $L$-functions.
Mahler measure, motivic regulators and Dirichlet $L$-values
本文通过研究动机上同调和狄利克雷L函数导数的关系，建立了马勒测度与调节器之间的公式，并在特定假设下推导了单个L值的恒等式。
In this series, we investigate the calculation of mean values of derivatives of Dirichlet $L$-functions in function fields using the analogue of the approximate functional equation and the Riemann Hypothesis for curves over finite fields. The present paper generalizes the results obtained in the first paper. For $μ\geq1$ an integer, we compute the mean value of the $μ$-th derivative of quadratic Dirichlet $L$-functions over the rational function field. We obtain the full polynomial in the asymptotic formulae for these mean values where we can see the arithmetic dependence of the lower order te…
A few years ago, the concept of a D-analogue was introduced as a Dirichlet series analogue for the already known and well researched hypergeometric q-series. The D-analogue of the q-Dixon sum is given here, in the context of seeing a direct comparison with the old Dixon theorem and the q-analogue.
We use the $q$-analogue of van der Corput's method to estimate short character sums to smooth moduli. If $χ$ is a primitive Dirichlet character modulo a squarefree, $q^δ$-smooth integer $q$ we show that $$L(\frac12,χ)\ll_εq^{\frac{27}{164}+O(δ)+ε}.$$
Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new conditional results on exponential sums with Hecke eigenvalues and squares of Hecke eigenvalues over primes. We employ these estimates to improve our recent result on squares of Hecke eigenvalues at Piatetski-Shapiro primes under the Riemann Hypothesis for symmetric square L-functions for Hecke eigenforms.
On a sum of the error term of the Dirichlet divisor function over primes
本文研究素数上Dirichlet除数函数误差项平方的求和问题，并证明了该和式存在渐近公式。
Let $q\ge3$ be an integer, $χ$ denote a Dirichlet character modulo $q$, for any real number $a\ge 0$, we define the generalized Dirichlet $L$-functions $$ L(s,χ,a)=\sum_{n=1}^{\infty}\frac{χ(n)}{(n+a)^s}, $$ where $s=σ+it$ with $σ>1$ and $t$ both real. It can be extended to all $s$ by analytic continuation. For any integer $m$, the famous Gauss sum $G(m,χ)$ is defined as follows: $$G(m,χ)=\sum_{a=1}^{q}χ(a)e\left(\frac{am}{q}\right), $$ where $e(y)=e^{2πiy}$. The main purpose of this paper is to use the analytic method to study the mean value properties of the generalized Dirichlet $L$-functio…
We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation formulae. Our calculation is novel in that the twisted sum can be expressed in terms of the average itself, and also that it sees the derivation of various new summation formulae in the setting of prime-power modulus. One consequence, as we explain, is to show the analytic continuation and additive summation formulae for hyper-Kloosterman Dirichlet series ass…
Moments of quadratic Dirichlet character sums
本文研究二次狄利克雷特征和的高次幂矩，利用多变量 Tauberian 定理得到渐近行为，并给出 Jutila 猜想中对数因子指数的下界及 L-函数平移矩加权平均的下界。
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。
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。
Moments of Higher Logarithmic Derivatives and Exceptional Zeros in Cyclotomic Fields
研究分圆域中与狄利克雷\(L\)函数相关的高阶对数导数\(\mathcal{L}^{(r)}(1,\chi)\)，通过证明算术公式和建立矩渐近性，得出足够大导体的素分圆域的第二李系数为正，排除斯塔克型无零区域中可能的例外零点。
We study the Dirichlet series $F_b(s)=\sum_{n=1}^\infty d_b(n)n^{-s}$, where $d_b(n)$ is the sum of the base-$b$ digits of the integer $n$, and $G_b(s)=\sum_{n=1}^\infty S_b(n)n^{-s}$, where $S_b(n)=\sum_{m=1}^{n-1}d_b(m)$ is the summatory function of $d_b(n)$. We show that $F_b(s)$ and $G_b(s)$ have continuations to the plane $\mathbb{C}$ as meromorphic functions of order at least 2, determine the locations of all poles, and give explicit formulas for the residues at the poles. We give a continuous interpolation of the sum-of-digits functions $d_b$ and $S_b$ to non-integer bases using a formu…
We attach a certain $n \times n$ matrix $A_n$ to the Dirichlet series $L(s)=\sum_{k=1}^{\infty}a_k k^{-s}$. We study the determinant, characteristic polynomial, eigenvalues, and eigenvectors of these matrices. The determinant of $A_n$ can be understood as a weighted sum of the first $n$ coefficients of the Dirichlet series $L(s)^{-1}$. We give an interpretation of the partial sum of a Dirichlet series as a product of eigenvalues. In a special case, the determinant of $A_n$ is the sum of the Möbius function. We disprove a conjecture of Barrett and Jarvis regarding the eigenvalues of $A_n$.
Distribution of sums involving Dirichlet characters over the $k$-free integers
在广义黎曼假设和Riemann zeta函数（或Dirichlet $L$-函数）负离散矩有界条件下，证明了$k$-无平方因子整数上归一化部分和$x^{-\frac{1}{2k}}\sum_{n\leq x}f(n)$的对数极限分布的存在性，并加强了关于这些部分和精确量级的猜想。
A method of estimating sums of multiplicative functions braided with Dirichlet characters is demonstrated, leading to a taxonomy of the characters for which such sums are large.
We have established novel integral representations of the Riemann zeta-function and Dirichlet eta-function based on powers of trigonometric functions and digamma function, and then use these representations to find close forms of Laurent series expansions of these same powers of trigonometric functions and digamma function. The so obtained series can be used to find numerous summation rules for certain values of the Riemann zeta and related functions and numbers, such as e.g. Bernoulli and Euler numbers.
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.
Fibonacci, Dirichlet, and Gauss in a single sum
研究分数部分和\(\sum_{k = 1}^{n}\{F_n/F_k\}\)，其渐近行为因\(n\)奇偶性而异，奇数\(n\)对应高斯圆问题，偶数\(n\)对应狄利克雷除数问题，还证明了二阶递推数列类似公式。
In this paper we study sums of Dirichlet series whose coefficients are terms of the Thue-Morse sequence and variations thereof. We find closed-form expressions for such sums in terms of known constants and functions including the Riemann zeta function and the Dirichlet eta function using elementary methods.
We study the double character sum $\sum\limits_{\substack{m\leq X,\\m\odd}}\sum\limits_{\substack{n\leq Y,\\n\odd}}\leg mn$ and its smoothly weighted counterpart. An asymptotic formula with power saving error term was obtained by Conrey, Farmer and Soundararajan by applying the Poisson summation formula. The result is interesting, because the main term involves a non-smooth function. In this paper, we apply the inverse Mellin transform twice and study the resulting double integral that involves a double Dirichlet series. This method has two advantages -- it leads to a better error term, and th…
We produce a new proof of the reciprocity law for the twisted second moment of Dirichlet L-functions that was recently proved by Conrey. Our method is to analyze certain two-variable sums where the variables satisfy a linear congruence. We show that these sums satisfy an elegant reciprocity formula. In the case that the modulus is prime, these sums are closely related to the twisted second moment, and the reciprocity formula for these sums implies Conrey's reciprocity formula. We also extend the range of uniformity of Conrey's formula.
Distribution of values of higher derivatives of $L'(s,χ)/L(s,χ)$
研究狄利克雷L函数对数导数的一阶导数的值分布理论，推广相关结果，探讨向高阶导数推广的障碍。
Zeros of $L$-functions and large partial sums of Dirichlet coefficients
该研究针对满足弱广义Ramanujan猜想的$L$-函数，证明其Dirichlet系数的大偏和会排斥低零点远离临界线，拓展并定量改进了前人相关研究成果。
Simple and accurate approximations to Dirichlet $L$-functions
该研究将文献[10]的求积方法推广到本原特征相关的Dirichlet L-函数，提出结合有限Dirichlet和与可预计算系数的修正项的近似方法，经实验验证精度高，还可结合直接求和与欧拉-麦克劳林求和实现复平面上的L(s,χ)计算。
We study the moments of $\mbox{Tr}(Θ_χ)$ as $χ$ runs over Dirichlet characters defined over $\mathbb{F}_q[T]$ of fixed order $r$. In particular, we show that after an appropriate normalization, the $q$-limit of the power sum moments behave like the power sum moments of the group of unitary matrices multiplied by a weight function.
The Dirichlet characters of reduced residue systems modulo m are tabulated for moduli m <= 195. The associated L-series are tabulated for m <= 14 and small positive integer argument s accurate to 10^(-50), their first derivatives for m <= 6. Restricted summation over primes only defines Dirichlet Prime L-functions which lead to Euler products (Prime Zeta Modulo functions). Both are materialized over similar ranges of moduli and arguments. Formulas and numerical techniques are well known; the aim is to provide direct access to reference values.
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 field. This is done by utilizing the functional equation of the $L$-functions and the functional equation relating the two multiple Dirichlet series. We also point out a very simple correspondence between these series and their $p$-parts.
The Laurent Stieltjes constants $γ_n(χ)$ are, up to a trivial coefficient, the coefficients of the Laurent expansion of the usual Dirichlet $L$-series: when $χ$ is non principal, $(-1)^nγ_n(χ)$ is simply the value of the $n$-th derivative of $L(s,χ)$ at $s=1$. In this paper, we give an approximation of the Dirichlet L-functions in the neighborhood of $s=1$ by a short Taylor polynomial. We also prove that the Riemann zeta function $ζ(s)$ has no zeros in the region $|s-1|\leq 2.2093,$ with $0\leq \Re{(s)}\leq 1.$
In this rather computational paper, we determine certain representation numbers of ideals in real quadratic number fields explicitly in order to obtain a representation of the associated Dirichlet series in terms of Dirichlet L-functions and a generalized divisor sum. A direct and important consequence is that the Dirichlet series has a meromorphic continuation to the whole complex plane and a simple pole at $s=2$ whose residue can be made explicit in terms of the Dirichlet L-functions and the generalized divisor sum.
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…
Special values of spectral zeta functions of graphs and Dirichlet L-functions
本文通过研究单位圆上一阶自伴微分算子的谱zeta函数，建立了一个组合导数公式，从而揭示了Dirichlet L-函数特殊值与圈图谱zeta函数特殊值之间的自然相互确定关系。
For certain sequences $A$ of positive integers with missing $g$-adic digits, the Dirichlet series $F_A(s) = \sum_{a\in A} a^{-s}$ has abscissa of convergence $σ_c < 1$. The number $σ_c$ is computed. This generalizes and strengthens a classical theorem of Kempner on the convergence of the sum of the reciprocals of a sequence of integers with missing decimal digits.
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.
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…
Modified Dirichlet character sums over the $k$-free integers
本文研究 $k$-自由整数上 $\pm 1$ 乘法函数部分和的抵消量，在二次狄利克雷 $L$-函数黎曼假设下，对修正二次狄利克雷特征证明了 $x^{1/(k+1)}$ 的抵消，改进了已有条件结果。
Let $k$ be a cubic field. We give an explicit formula for the Dirichlet series $\sum_K|\Disc(K)|^{-s}$, where the sum is over isomorphism classes of all quartic fields whose cubic resolvent field is isomorphic to $k$. Our work is a sequel to an unpublished preprint of Cohen, Diaz y Diaz, and Olivier, and we include complete proofs of their results so as not to rely on unpublished work. This is a companion to a previous paper where we compute the Dirichlet series associated to cubic fields having a given quadratic resolvent.
Omega theorems for logarithmic derivatives of zeta and $L$-functions
本文为黎曼zeta函数和Dirichlet L-函数的对数导数建立了新的Omega定理，改进了Landau、Bohr-Landau及Lamzouri的先前结果。
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。
The paper considers a method for converting a divergent Dirichlet series into a convergent Dirichlet series by directly converting the coefficients of the original series $1\rightarrowδ_{n}(s)$ for the Riemann Zeta function. In the first part of the paper, we study the properties of the coefficients $δ^*_n$ of a finite Dirichlet series for approximating the Riemann Zeta function on the interval $Δ{H}$. In general, the coefficients $δ^*_n$ of a finite Dirichlet series are complex numbers. The dependence of the coefficients $δ^*_n$ of a finite Dirichlet series on the ordinal number of the coeffi…
We compute the second moment in the family of quadratic Dirichlet $L$-functions with prime conductors over $\mathbb{F}_q[x]$ when the degree of the discriminant goes to infinity, obtaining one of the lower order terms. We also obtain an asymptotic formula with the leading order term for the mean value of the derivatives of $L$-functions associated to quadratic twists of a fixed elliptic curve over $\mathbb{F}_q(t)$ by monic irreducible polynomials, which allows us to show that there exists a monic irreducible polynomial such that the analytic rank of the corresponding twisted elliptic curve is…
We prove an asymptotic formula for the mean-square average of $L$- functions associated to subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums ${\cal A}(p,d)$ recently introduced by E. Elma. We obtain an asymptotic formula for ${\cal A}(p,d)$ which holds true for any divisor $d$ of $p-1$ removing previous restrictions on the size of $d$. This anwers a question raised in Elma's paper. Our proof relies both on estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application we dedu…
The Ratios conjecture for real Dirichlet characters and multiple Dirichlet series
本文在广义黎曼假设下，利用多重狄利克雷级数证明实狄利克雷特征族的单平移L函数比率猜想，扩展至非本原特征可增大平移范围，进而得到平移对数导数和的渐近公式。
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 error terms gives information on the the location of zeros of L-functions and possible Siegel zeros. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.
Given $c,$ a positive integer, we give an explicit formula and an asymptotic formula for \[ \sumχ(c)|L(1,\,χ)|^{2}, \] where $χ$ is the non-trivial Dirichlet character mod $f$ with $f>c.$
For a positive irrational number $α,$ we study the ordinary Dirichlet series $ζ_α(s) = \sum\limits_{n\geq1} \lfloorαn\rfloor^{-s}$ and $S_α(s) = \sum\limits_{n\geq1} (\left\lceilαn\right\rceil - \left\lceil α(n-1)\right\rceil){n^{-s}}.$ We prove relations between them and $J_{\boldsymbolα}(s)=\sum\limits_{n\geq1}\left(\lbraceαn\rbrace-\frac{1}{2}\right)n^{-s}.$ Motivated by the previous work of Hardy and Littlewood, Hecke and others regarding $J_{\boldsymbolα},$ we show that $ζ_α$ and $S_α$ can be continued analytically beyond the imaginary axis except for a simple pole at $s=1.$ Based on the …
We provide a general theorem for evaluating trigonometric Dirichlet series of the form $\sum_{n \geq 1} \frac{f (πn τ)}{n^s}$, where $f$ is an arbitrary product of the elementary trigonometric functions, $τ$ a real quadratic irrationality and $s$ an integer of the appropriate parity. This unifies a number of evaluations considered by many authors, including Lerch, Ramanujan and Berndt. Our approach is based on relating the series to combinations of derivatives of Eichler integrals and polylogarithms.
On the regularized products of some Dirichlet series
本文证明了某些Dirichlet级数的正则化行列式具有乘性，并以此为应用推广了Lerch公式和Euler关于平方倒数和的公式，同时恢复了Kurokawa与Wakayama的结果。
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