Subconvexity for the Riemann zeta-function and the divisor problem

Moments of zeta and correlations of divisor-sums: III

In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper is concerned with the precise input of the conjectural formula for the classical shifted convolution problem for divisor sums so as to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of length up to $T^2$ with divisor functions as coefficients.

除数函数的极大阶与黎曼ζ函数的零点

Maximal order for divisor functions and zeros of the Riemann zeta-function

本文将黎曼ζ函数的无零点区域与除数幂和函数的极大阶关联，给出了用除数的1/2次幂和函数及黎曼ζ函数部分欧拉乘积表示的黎曼假设等价条件。

黎曼zeta函数与除数函数的精确指数和恒等式

The Riemann zeta function and exact exponential sum identities of divisor functions

本文证明了一个计算两个移位黎曼zeta函数乘积的显式积分公式，由此推导出涉及除数函数的无穷族精确指数和恒等式，并提出了无需算术技巧的矩阵求逆计算除数函数的推测性方法。

Moments of zeta and correlations of divisor-sums: I

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.

Moments of zeta and correlations of divisor-sums: IV

In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper begins the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of length up to $T^3$ with divisor functions as coefficients.

Moments of zeta and correlations of divisor-sums: V

In this series of papers we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper completes the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of arbitrary length with divisor functions as coefficients.

Averages of ratios of the Riemann zeta-function and correlations of divisor sums

We establish a connection between the ratios conjecture for the Riemann zeta-function and a conjecture concerning correlations of convolutions of Möbius and divisor functions. Specifically, we prove that the ratios conjecture and an arithmetic correlations conjecture imply the same result. This provides new support for the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe. Our main theorem generalises a recent calculation pertaining to the special case of two-over-two ratios.

Moments of zeta and correlations of divisor-sums: stratification and Vandermonde integrals

We refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta-function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.

The algebra of generating functions for multiple divisor sums and applications to multiple zeta values

We study the algebra MD of generating function for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in q with coefficients in Q arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra MD is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in MD. The (quasi-)modular forms for the full modular group Sl_2(Z) constitute a sub-algebra of MD this also yields linear relations in MD. Generating functions of mul…

On some upper bounds for the zeta-function and the Dirichlet divisor problem

Let $d(n)$ be the number of divisors of $n$, let $$ Δ(x) := \sum_{n\le x}d(n) - x(\log x + 2γ-1) $$ denote the error term in the classical Dirichlet divisor problem, and let $ζ(s)$ denote the Riemann zeta-function. Several upper bounds for integrals of the type $$ \int_0^TΔ^k(t)|ζ(1/2+it)|^{2m}dt \qquad(k,m\in\Bbb N) $$ are given. This complements the results of the paper Ivić-Zhai [Indag. Math. 2015], where asymptotic formulas for $2\le k \le 8,m =1$ were established for the above integral.

Numerical computations on the zeros of the Euler double zeta-function II

We study the behavior of zero-divisors of the double zeta-function $ζ_2(s_1,s_2)$. In our former paper \cite{MatSho14} we studied the case $s_1=s_2$, but in the present paper we consider the more general two variable situation. We carry out numerical computations in order to trace the behavior of zero-divisors. We observe that some divisors approach the points with $s_2=0$, while other divisors approach the points which are solutions of $ζ(s_2)=1$.

The sixth moment of the Riemann zeta function and ternary additive divisor sums

Hardy and Littlewood initiated the study of the $2k$-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. Since then no other moments have been asymptotically evaluated. In this article we study the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula with power savings error term for the sixth moment of th…

On the Selberg integral of the $k-$divisor function and the $2k-$th moment of the Riemann zeta function

We deeply appreciate the papers of Ivić on the links between the $2k-$th moments of the Riemann zeta function and, say, $d_k$, the $k-$divisor function. More specifically, both the one bounding the $2k-$th moment with a simple average of correlations of the $d_k$ (Palanga 1996 Conference Proceedings) and the more recent (arXiv:0708.1601v2 to appear on JTNB), which bounds the Selberg integral of $d_k$ applying the $2k-$th moment of the zeta. Building on the former paper, we apply an elementary approach (based on arithmetic averages) in order to get information on the reverse link to the second …

On the Riemann zeta-function and the divisor problem IV

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 it is proved that $$ \int_0^T|E^*(t)|^3dt \ll_εT^{3/2+ε}, $$ which is (up to `$ε$' best possible) and $ζ(1/2+it) \ll_εt^{ρ/2+ε}$ if $E^*(t) \ll_εt^{ρ+ε}$.

On the Riemann zeta-function and the divisor problem II

First part of this paper was published in CEJM (2)(4) (2004), 1-15. It is proved now that $$ \int_0^T|E^*(t)|^5{\rm d}t \ll_εT^{2+ε}. $$ Here $$ E^*(t) = E(t) - 2πΔ^*(t/2π), Δ^*(x) = - Δ(x) +2Δ(2x) - {1\over2}Δ(4x), $$ where $E(t)$ is the error term in the mean square formula for $|ζ(1/2+it)|$ and $Δ(x)$ is the error term in the Dirichlet divisor problem. It is also shown how bounds for moments of $|E^*(t)|$ lead to bounds for moments of $|ζ(1/2+it)|$.

On the mean square of the zeta-function and the divisor problem

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 the asymptotic formula $$ \int_0^T (E^*(t))^2 {\rm d} t = T^{4/3}P_3(\log T) + O_ε(T^{7/6+ε}), $$ where $P_3$ is a polynomial of degree three in $\log T$ with positive leading coefficient. The exponent 7/6 in the error term is the limit of the method.

On the Riemann zeta-function and the divisor problem III

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 $Δ^*(x) = -Δ(x) + 2Δ(2x) - {1\over2}Δ(4x)$ and we set $\int_0^T E^*(t) dt = 3πT/4 + R(T)$, then we obtain $$ R(T) = O_ε(T^{593/912+ε}), \int_0^TR^4(t) dt \ll_εT^{3+ε}, $$ and $$ \int_0^TR^2(t) dt = T^2P_3(\log T) + O_ε(T^{11/6+ε}), $$ where $P_3(y)$ is a cubic polynomial in $y$ with positive leading coefficient.

A quadratic divisor problem and moments of the Riemann zeta-function

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…

On the divisor function and the Riemann zeta-function in short intervals

We obtain, for $T^ε\le U=U(T)\le T^{1/2-ε}$, asymptotic formulas for $$ \int_T^{2T}(E(t+U) - E(t))^2 dt,\quad \int_T^{2T}(Δ(t+U) - Δ(t))^2 dt, $$ where $Δ(x)$ is the error term in the classical divisor problem, and $E(T)$ is the error term in the mean square formula for $|ζ(1/2+it)|$. Upper bounds of the form $O_ε(T^{1+ε}U^2)$ for the above integrals with biquadrates instead of square are shown to hold for $T^{3/8} \le U =U(T) \ll T^{1/2}$. The connection between the moments of $E(t+U) - E(t)$ and $|ζ(1/2+it)|$ is also given. Generalizations to some other number-theoretic error terms are discu…

关于黎曼ζ函数的林德勒夫假设与皮尔茨除数问题

On the Lindelöf Hypothesis for the Riemann Zeta function and Piltz divisor problem

本文通过给出临界带内ζᵏ(s)的傅里叶展开推导林德勒夫假设的充要条件，并针对皮尔茨除数问题的误差项Δₖ(x)，得到了几乎所有x≥1时其与傅里叶系数及拉盖尔多项式相关的极限表达式。

On some mean value results for the zeta-function and a divisor problem II

Let $d(n)$ be the number of divisors of $n$, let $γ$ denote Euler's constant and $$ Δ(x) := \sum_{n\le x}d(n) - x(\log x + 2γ-1) $$ denote the error term in the classical Dirichlet divisor problem, and let $ζ(s)$ denote the Riemann zeta-function. It is shown that $$ \int_0^TΔ(t)|ζ(1/2+it)|^2\,dt \ll T(\log T)^{4}. $$ Further, if $2\le k\le 8$ is a fixed integer, then we prove the asymptotic formula $$ \int_1^{T}Δ^{k}(t)|ζ(1/2+it)|^2\,dt=c_1(k)T^{1+\frac k4}\log T+ c_2(k)T^{1+\frac k4}+O_\varepsilon(T^{1+\frac k4-η_k+\varepsilon}), $$ where $c_1(k)$ and $c_2(k)$ are explicit constants, and wher…

Even powers of divisors and elliptic zeta values

We introduce and study "elliptic zeta values", a two-parameter deformation of the values of Riemann's zeta function at positive integers. They are essentially Taylor coefficients of the logarithm of the elliptic gamma function, and share the SL(3,Z) modular properties of this function. Elliptic zeta values at even integers are related to Eisenstein series and thus to sums of odd powers of divisors. The elliptic zeta values at odd integers can be expressed in terms of generating series of sums of even powers of divisors.

Multiple Divisor Functions and Multiple Zeta Values at Levle N

Multiple zeta values (MZVs) are generalizations of Riemann zeta values at positive integers to multiple variable setting. These values can be further generalized to level $N$ multiple polylog values by evaluating multiple polylogs at $N$-th roots of unity. In this paper, we consider another level $N$ generalization by restricting the indices in the iterated sums defining MZVs to congruences classes modulo $N$, which we call the MZVs at level $N$. The goals of this paper are two-fold. First, we shall lay down the theoretical foundations of these values such as their regularizations and double s…

On the Riemann zeta-function and the divisor problem

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…

On some mean value results for the zeta-function and a divisor problem

Let $Δ(x)$ denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be $Δ^*(x) = -Δ(x) + 2Δ(2x) - \frac{1}{2}Δ(4x)$. We show that $$ \int_T^{T+H}Δ^*\bigl(\frac{t}{2π}\bigr)|ζ(1/2+it)|^2dt \;\ll\; HT^{1/6}\log^{7/2}T \quad(T^{2/3+\varepsilon} \le H = H(T) \le T), $$ $$ \int_0^TΔ(t)|ζ(1/2+it)|^2dt \;\ll\; T^{9/8}(\log T)^{5/2},$$ and obtain asymptotic formulae for $$ \int_0^T{\Bigl(Δ^*\bigl(\frac{t}{2π}\bigr)\Bigr)}^2 |ζ(1/2+it)|^2dt,\quad \int_0^T{\Bigl(Δ^*\bigl(\frac{t}{2π}\bigr)\Bigr)}^3|ζ(1/2+it)|^2dt. $$ The importance of the $…

On the zeta function of divisors for projective varieties with higher rank divisor class group

Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of C_p. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case…

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。

黎曼 zeta 函数辐角与零点计数函数的改进估计

Improved estimates for the argument and zero-counting function of the Riemann zeta-function

本文通过引入临界带内若干 σ_k-线上 ζ(σ+it) 的新次凸界，改进了黎曼 zeta 函数零点计数函数 N(T) 的显式误差估计。

ζ(1/2+it)的一个新的改进显式估计

A new improved explicit estimate for $ζ\left( 1/2+it\right)$

本文通过改进的van der Corput方法和计算，提出了一种新的ζ(1/2+it)的显式估计结果。

Divisor Problem and an Analogue of Euler's Summation Formula

By simple elementary method,we obtain with ease,a highly simple expression for the remainder term of the divisor problem and use it to obtain an Euler-Maclaurin analogue of summation involving divisor function.We also obtain a relation connecting the remainder term of the divisor problem and the remainders of approximate functional equations for Riemann zeta function and its square,for positive values of arguments.

The eighth moment of the Riemann zeta function

In this article, we establish an asymptotic formula for the eighth moment of the Riemann zeta function, assuming the Riemann hypothesis and a quaternary additive divisor conjecture. This builds on the work of the first author on the sixth moment of the Riemann zeta function and work of Conrey-Gonek and Ivić. A key input is a sharp bound for a certain shifted moment of the Riemann zeta function, assuming the Riemann hypothesis.

Multiple finite Riemann zeta functions

Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some $q$-series identity for proving the zeta function has an Euler product and then, describe the location of zeros. We study further multi-variable and multi-parameter versions of the multiple finite Riemann zeta functions and their infinite counterparts in connection with symmetric polynomials and some arithmetic quantities called powerful numbers.

Pair correlation and twin primes revisited

We use the methods developed in our papers on moments and divisor correlations to derive heuristically the conjectural ratios formula for two zetas over two zetas.

利用共振法得到的除数问题与圆问题的Ω结果

Omega Results for The Divisor and Circle Problems Using The Resonance Method

该研究将共振法用于获取带正系数的一般指数和的大值，改进了数论中多个经典问题的Ω界，包括除数问题、圆问题及zeta函数和L函数的均方误差问题。

二平方四次幂上的除数函数

The divisor function along sums of two biquadrates

本文研究了二平方四次幂上除数函数的渐进行为，采用二维delta方法和GL₂自守形式分析，改进了Daniel的成果。

黎曼ζ函数的无零点区域

Zero-free regions for the Riemann zeta function

本文利用作者关于ζ(σ+it)增长的显式界及一种关联函数竖直线增长与竖条零点加权计数的Jensen公式，改进了黎曼ζ函数无零点区域的Vinogradov-Korobov型显式界（含大高度t和所有t的情况）。

Dirichlet divisor problem on Gaussian integers

We improve existing estimates of moments of the Riemann zeta function. As a consequence, we are able to derive new estimates for the asymptotic behaviour of $\sum_{N α\le x} \mathfrak{t}_k(α)$, where $N$ stands for the norm of a complex number and $\mathfrak{t}_k$ is the $k$-dimensional divisor function on Gaussian integers.

Zeta functions of nondegenerate hypersurfaces in toric varieties via controlled reduction in $p$-adic cohomology

We give an interim report on some improvements and generalizations of the Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time in $p$. These are illustrated with a number of examples including K3 surfaces, Calabi-Yau threefolds, and a cubic fourfold. The latter example is a non-special cubic fourfold appearing in the Ranestad-Voisin coplanar divisor on moduli space; this verifies that the coplanar divisor is not a Noether-Lefschetz divisor in the sense of Hassett.

Class number and regulator computation in cubic function fields

We present computational results on the divisor class number and the regulator of a cubic function field over a large base field. The underlying method is based on approximations of the Euler product representation of the zeta function of such a field. We give details on the implementation for purely cubic function fields of signatures $(3,1)$ and $(1, 1; 1, 2)$, operating in the ideal class group and infrastructure of the function field, respectively. Our implementation provides numerical evidence of the computational effectiveness of this algorithm. With the exception of special cases, such …

On a hybrid fourth moment involving the Riemann zeta-function

We provide explicit ranges for $σ$ for which the asymptotic formula \begin{equation*} \int_0^T|ζ(1/2+it)|^4|ζ(σ+it)|^{2j}dt \;\sim\; T\sum_{k=0}^4a_{k,j}(σ)\log^k T \quad(j\in\mathbb N) \end{equation*} holds as $T\rightarrow \infty$, when $1\leq j \leq 6$, where $ζ(s)$ is the Riemann zeta-function. The obtained ranges improve on an earlier result of the authors [Annales Univ. Sci. Budapest., Sect. Comp. {\bf38}(2012), 233-244]. An application to a divisor problem is also given

Bounds and Conjectures for additive divisor sums

Additive divisor sums play a prominent role in the theory of the moments of the Riemann zeta function. There is a long history of determining sharp asymptotic formula for the shifted convolution sum of the ordinary divisor function. In recent years, it has emerged that a sharp asymptotic formula for the shifted convolution sum of the triple divisor function would be useful in evaluating the sixth moment of the Riemann zeta function. In this article, we study $D_{k,\ell}(x) = \sum_{n \le x} τ_k(n) τ_{\ell}(n+h)$ where $τ_k$ and $τ_{\ell}$ are the $k$-th and $\ell$-th divisor functions. The main…

Dirichlet series for complex powers of the Riemann zeta function

To obtain the Dirichlet series for complex powers of the Riemann zeta function, we define and study the basic properties of a sequence of polynomials that, used as coefficients of the respective terms of the Dirichlet series of the Riemann zeta function in the half plane $x > 1$, produces the required exponential function. Unlike the method described in ([4], p.~278), which requires more advanced knowledge of the relationships between Dirichlet series and multiplicative arithmetic functions, our approach only needs mathematical induction on the total number of prime divisors of $n$, the Dirich…

A mean value theorem for orders of degree zero divisor class groups of quadratic extensions over a function field

Let $k$ be a function field of one variable over a finite field with the characteristic not equal to two. In this paper, we consider the prehomogeneous representation of the space of binary quadratic forms over $k$. We have two main results. The first result is on the principal part of the global zeta function associated with the prehomogeneous vector space. The second result is on a mean value theorem for degree zero divisor class groups of quadratic extensions over $k$, which is a consequence of the first one.

重探 zeta 函数的均方、圆问题与除数问题

Mean square of zeta function, circle problem and divisor problem revisited

本文围绕 zeta 函数均方、圆问题与除数问题，通过改进 [BW17] 第 4 节相关方法，给出其定理 3 的新结果。

Multiple Dirichlet series and moments of zeta and L-functions

This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic L-series. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of central values of Dirichlet L-series.…

On some mean square estimates for the zeta-function in short intervals

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 $Δ^*(x) = -Δ(x) + 2Δ(2x) - 1/2Δ(4x)$ and we set $\int_0^T E^*(t)\,dt = 3πT/4 + R(T)$, then we obtain $$ \int_T^{T+H}(E^*(t))^2\,dt \gg HT^{1/3}\log^3T $$ and $$ HT\log^3T \ll \int_T^{T+H}R^2(t)\,dt \ll HT\log^3T, $$ for $T^{2/3+ε}\le H \le T$.

On the modified Selberg integral of the three-divisor function $d_3$

We prove a non-trivial result for the,say,modified Selberg integral $\modSel_3(N,h)$, of the divisor function $d_3(n):= \sum_{a}\sum_{b}\sum_{c, abc=n}1$; this integral is a slight modification of the corresponding Selberg integral, that gives the expected value of the function in short intervals. We get, in fact, $\modSel_3(N,h)\ll Nh^2L^2$, where $L:=\log N$; furthermore, as a byproduct, we obtain indications on the way in which it may be proved the weak sixth moment of the Riemann zeta function.(This was OLD abstract)

根系zeta函数的通用极除子和旗标留数

Generic polar divisors and flag residues for root-system zeta functions

研究根系zeta函数，证明特定超平面是极除子，推导递归旗标留数等公式，恢复经典奇异数据及相关留数函数，确定$B_3$和$C_3$的一些性质，如双极点位置等。

Motivic Zeta Functions on $\mathds{Q}$-Gorenstein Varieties

We study motivic zeta functions for $\mathds{Q}$-divisors in a $\mathds{Q}$-Gorenstein variety. By using a toric partial resolution of singularities we reduce this study to the local case of two normal crossing divisors where the ambient space is an abelian quotient singularity. For the latter we provide a closed formula which is worked out directly on the quotient singular variety. As a first application we provide a family of surface singularities where the use of weighted blow-ups reduces the set of candidate poles drastically. We also present an example of a quotient singularity under the …

The Divisor Matrix, Dirichlet Series and SL(2,Z)

A representation of SL(2,Z) by integer matrices acting on the space of analytic ordinary Dirichlet series is constructed, in which the standard unipotent element acts as multiplication by the Riemann zeta function. It is then shown that the Dirichlet series in the orbit of the zeta function are related to it by algebraic equations.

Motohashi's fourth moment identity for non-archimedean test functions and applications

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 …

Approximate Functional Equation for the Product of Functions and Divisor Problem

For functions defined via Dirichlet/generalized Dirichlet series in some half planes of the complex plane, we give a new simple elementary approach to obtain an Approximate Functional Equation(AFE for short) for the product of functions with explicit remainder term. We do this,using a highly generalized identity following Motohashi's approach,wherein he uses a generalisation of Dirichlet's device.We use our method to give an AFE for the product of two Dirichlet L-series corresponding to primitive Dirichlet characters with hitherto unknown estimate for its remainder term.We also consideer AFEs …

On some mean value results for the zeta-function in short intervals

Let $Δ(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote 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 $Δ^*(x) := -Δ(x) + 2Δ(2x) - \frac{1}{2}Δ(4x)$ and $\int_0^T E^*(t)\,dt = \frac{3}{4}πT + R(T)$, then we obtain a number of results involving the moments of $|ζ(1/2+it)|$ in short intervals, by connecting them to the moments of $E^*(T)$ and $R(T)$ in short intervals. Upper bounds and asymptotic formulas for integrals of the form $$ \int_T^{2T}\left(\int_{t-H}^{t+H}|ζ(1/2+iu)|^2\,du\right)^…

Superzeta functions, regularized products, and the Selberg zeta function on hyperbolic manifolds with cusps

Let $Λ= \{λ_{k}\}$ denote a sequence of complex numbers and assume that that the counting function $#\{λ_{k} \in Λ: | λ_{k}| < T\} =O(T^{n})$ for some integer $n$. From Hadamard's theorem, we can construct an entire function $f$ of order at most $n$ such that $Λ$ is the divisor $f$. In this article we prove, under reasonably general conditions, that the superzeta function $\Z_{f}(s,z)$ associated to $Λ$ admits a meromorphic continuation. Furthermore, we describe the relation between the regularized product of the sequence $z-Λ$ and the function $f$ as constructed as a Weierstrass product. In t…