Dirichlet Divisor Research Articles

A geometry where everything is better than nice

We present a riemannian structure on the disk that has a remarkably rich structure. Geodesics are hypocycloids and the (negative of the) laplacian has integer spectrum with multiplicity the Dirichlet divisor function. Eigenfunctions of the laplacian are orthogonal polynomials naturally suited to the analysis of acoustic scattering in layered media.

On the mean of the shifted error term in the theory of the Dirichlet divisor problem

Let $0\lt \alpha \lt 1$. We show that Bernoulli polynomials appear in the difference $\sum _{n \leq x}\Delta ^j(n+\alpha )-\int _1^x \Delta ^j(t)\,dt$ for $j=1, \ldots , 4$. As a corollary of this fact, we get better approximations of $\int _1^x \Delta ^j(t)\,dt$ by using zeros of Bernoulli polynomials. For $j=1,2$, we give some interpretation of this fact by means of Dirichlet series with the coefficients $\Delta ^j(n+\alpha )$.

On the Moments of the Function E*(t)

Let <img src=image/abs1.png></img> denote the error term in the Dirichlet divisor problem, and <img src=image/abs2.png></img><i> </i>the error term in the asymptotic formula for the mean square of <img src=image/abs3.png></img> If<i> </i><img src=image/abs4.png></img><i> </i>with <img src=image/abs5.png></img> then we discuss bounds for third, fourth and fifth power moment of <img src=image/abs6.png></img><i> </i>We also prove that <img src=image/abs7.png></img><i> </i>always changes sign in <img src=image/abs8.png></img> for<i> </i><img src=image/abs9.png></img> and obtain (conditionally) the existence of its large positive, or small negative values.

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

Let ?(x) denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be ?*(x) = -?(x) + 2?(2x)-1/2?(4x). We show that ?T+H,T ?*(t/2?)|?(1/2+it)|2dt&lt;&lt; HT1/6log7/2 T (T2/3+? ? H = H(T) ? T), ?T,0 ?(t)|?(1/2+it)|2dt &lt;&lt; T9/8(log T)5/2, and obtain asymptotic formulae for ?T,0 (?*(t/2?))2|?( 1/2+it)|2 dt, ?T0 (?*(t/2?))3|?(1/+it)|2 dt. The importance of the ?*-function comes from the fact that it is the analogue of E(T), the error term in the mean square formula for |?(1/2+it)|2. We also show, if E*(T) = E(T)-2??*(T/(2?)), ?T0 E*(t)Ej(t)|?(1/2+it)|2 dt &lt;&lt; j,? T7/6+j/4+? (j=1,2,3).

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 Δ(x)≔∑n⩽xd(n)−x(logx+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 ∫0TΔ(t)|ζ(12+it)|2dt≪T(logT)4. Further, if 2⩽k⩽8 is a fixed integer, then we prove the asymptotic formula ∫1TΔk(t)|ζ(12+it)|2dt=c1(k)T1+k4logT+c2(k)T1+k4+Oε(T1+k4−ηk+ε), where c1(k) and c2(k) are explicit constants, and where η2=3/20,η3=η4=1/10,η5=3/80,η6=35/4742,η7=17/6312,η8=8/9433. The results depend on the power moments of Δ(t) and E(T), the classical error term in the asymptotic formula for the mean square of |ζ(12+it)|.

On the differences between two kinds of mean value formulas of number-theoretic error terms

In this paper, we shall consider differences between the "discrete" and the "continuous" mean value formulas for any number-theoretic error term. We first derive new type formulas of these differences, which are independent from the periodic Bernoulli function. As a direct consequence of this result, we shall give the alternative proofs of previous results in the case of the Dirichlet divisor and the circle problems, and improve the previous results of the differences in higher power cases. Furthermore, we apply our present result to the case of Rankin–Selberg's series.

Weyl asymptotics for tensor products of operators and Dirichlet divisors

We study the counting function of the eigenvalues for tensor products of operators, and their perturbations, in the context of Shubin classes and closed manifolds. We emphasize connections with problems of analytic number theory, concerning in particular generalized Dirichlet divisor functions.

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

Let $\varDelta(x)$ 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 $|\zeta(1/2+it)|$. If $E^*(t) := E(t) - 2\pi\varDelta^*(t/(2\pi))$ with $\varDelta^*(x) = -\v

The Dirichlet divisor problem, traces and determinants for complex powers of the twisted bi-Laplacian

Estimating the counting function for the eigenvalues of the twisted bi-Laplacian leads to the Dirichlet divisor problem, which is then used to compute the trace of the heat semigroup and the Dixmier trace of the inverse of the twisted bi-Laplacian. The zeta function regularizations of the traces and determinants of complex powers of the twisted bi-Laplacian are computed. A formula for the zeta function regularizations of determinants of heat semigroups of complex powers of the twisted bi-Laplacian is given.

On the Dirichlet divisor problem in short intervals

We present several new results involving Δ(x+U)−Δ(x), where U=o(x) and $$\varDelta(x):=\sum_{n\leq x}d(n)-x\log x-(2\gamma-1)x $$ is the error term in the classical Dirichlet divisor problem.

Voronoi–Nasim summation formulas and index transforms

Using L 2-theory of the Mellin and Fourier–Watson transformations, we relax Nasim’s conditions to prove the summation formula of Voronoi. It involves sums of the form , where d(n) is the number of divisors of n. These sums are related to the famous Dirichlet divisor problem of determining the asymptotic behaviour as of the sum . In particular, we generalize Koshliakov’s formula, which contains the modified Bessel function of zero-index f(x)=K 0(2πz x), (z is a parameter) on the modified Bessel function of an arbitrary complex index. Finally, we apply index transforms of the Kontorovich–Lebedev type to obtain a new class of summation formulas involving Dirichlet’s function~d(n).

Weyl asymptotics of bisingular operators and Dirichlet divisor problem

We consider a class of pseudodifferential operators, with crossed vector valued symbols, defined on the product of two closed manifolds. We study the asymptotic expansion of the counting function of positive selfadjoint operators in this class. Using a general Theorem of Aramaki, we can determine the first term of the asymptotic expansion of the counting function and, in a special case, we are able to find the second term. We give also some examples, emphasizing connections with problems of analytic number theory, in particular with Dirichlet divisor function.

EXPLICIT REPRESENTATIONS OF THE INTEGRAL CONTAINING THE ERROR TERM IN THE DIVISOR PROBLEM II

AbstractIn our previous paper [2], we derived an explicit representation of the integral ∫1∞t−θΔ(t)logjtdt by differentiation under the integral sign. Here, j is a fixed natural number, θ is a complex number with 1 &lt; θ ≤ 5/4 and Δ(x) denotes the error term in the Dirichlet divisor problem. In this paper, we shall reconsider the same formula by an alternative approach, which appeals to only the elementary integral formulas concerning the Riemann zeta- and periodic Bernoulli functions. We also study the corresponding formula in the case of the circle problem of Gauss.

A general class of Voronoi's and Koshliakov–Ramanujan's summation formulas involving d k (n)

By using the theory of the Mellin and Mellin convolution type transforms, we prove a general summation formula of Voronoi involving sums of the form ∑ d k (n)f(n), where d k (n), k=2, 3, …, d 2(n)≡ d(n) is the number of ways of expressing n as a product of k factors. These sums are related to the famous Dirichlet divisor problem of determining the asymptotic behaviour as x→∞ of the sum D k (x)=∑ n≤x d k (n). In particular, we generalize Koshliakov's formula and certain identities from Ramanujan's lost notebook to the case of hyper-Bessel functions and Jacobi elliptic theta functions. New examples of Voronoi's summation formulas involving Bessel, exponential functions and their products, which are based on a comprehensive Marichev's table of Mellin's transforms are given. The equivalence of these relations to the functional equation for the Riemann Zeta -function is discussed. An extension of the Koshliakov formula involving the Kontorovich–Lebedev transform is obtained.

A new proof of the Voronoï summation formula

We present a short alternative proof of the Voronoï summation formula which plays an important role in Dirichlet's divisor problem and has recently found an application in physics as a trace formula for a Schrödinger operator on a noncompact quantum graph.

New estimates of the remainder in an asymptotic formula in the multidimensional Dirichlet divisor problem

We obtain a new value of the Karatsuba constant in the multidimensional Dirichlet divisor problem. We also find a new value of the exponent of the main parameter in the estimate of the mean value of the remainder in a given asymptotics. The proof of the main statements is based on the derivation of a new estimate of the Carleson abscissa in the theory of the Riemann zeta function.

ON THE FIFTH-POWER MOMENT OF Δ(x)

Let Δ(x) denote the error term in the Dirichlet divisor problem. In this paper, we study the fifth-power moment of Δ(x) and prove that [Formula: see text] with δ5 = 3/80, which improves the previous results.

Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function

Let Δ(x) and E(t) denote respectively the remainder terms in the Dirichlet divisor problem and the mean square formula for the Riemann zeta-function on the critical line. This article is a survey of recent developments on the research of these famous error terms in number theory. These include upper bounds, Ω-results, sign changes, moments and distribution, etc. A few open problems are also discussed.

Analytic properties of Dirichlet series obtained from the error term in the Dirichlet divisor problem

Analytic properties of Dirichlet series obtained from the error term in the Dirichlet divisor problem

Dirichlet series obtained from the error term in the Dirichlet divisor problem

In this paper we consider the Dirichlet series obtained from the error term in the Dirichlet divisor problem. We shall prove their analytic continuations and determine the locations of poles. We shall also discuss the relation between our results and the conjecture of Lau and Tsang on the mean square estimate of the error term.