Dirichlet Divisor Research Articles

Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem

In this paper we study generalised prime systems for which the integer counting function N P ( x ) is asymptotically well behaved, in the sense that N P ( x ) = ρ x + O ( x β ) , where ρ is a positive constant and β < 1 2 . For such systems, the associated zeta function ζ P ( s ) is holomorphic for σ = R s > β ( s ≠ 1 ) . We prove that for β < σ < 1 2 , ∫ 0 T | ζ P ( σ + i t ) | 2 d t = Ω ( T 2 − 2 σ − ε ) for any ε > 0 , and also for ε = 0 for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in N k P ( x ) − Res s = 1 ( ζ P ( s ) k x s / s ) , which is O ( x θ ) for some θ < 1 . Letting α k denote the infimum of such θ, we show that α k ⩾ 1 2 − 1 2 k .

The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy

We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchywith complex-valued initial data and prove unique solvability globally intime for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inversealgebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy. The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to $(1+1)$-dimensional completely integrable soliton equations of differential-difference type.

On the mean square formula of the error term in the Dirichlet divisor problem

AbstractLet F(x) be the remainder term in the mean square formula of the error term Δ(t) in the Dirichlet divisor problem. We improve on the upper estimate of F(x) obtained by Preissmann around twenty years ago. The method is robust, which applies to the same problem for the error terms in the circle problem and the mean square formula of the Riemann zeta-function.

The algebro-geometric Toda hierarchy initial value problem for complex-valued initial data

We discuss the algebro-geometric initial value problem for the Toda hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Toda hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for generally non-self-adjoint Jacobi operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin equations), this yields the construction of global algebro-geometric solutions of the time-dependent Toda hierarchy. The inherent non-self-adjointness of the underlying Lax (i.e., Jacobi) operator associated with complex-valued coefficients for the Toda hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Toda hierarchy but applies generally to 1+1 -dimensional completely integrable discrete soliton equations.

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

Let ?(T) and E(T) be the error terms in the classical Dirichlet divisor problem and in the asymptotic formula for the mean square of the Riemann zeta function in the critical strip, respectively. We show that ?(T) and E(T) are asymptotic integral transforms of each other. We then use this integral representation of ?(T) to give a new proof of a result of M. Jutila.

On the higher moments of the error term in the divisor problem

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas for the integral of the cube and the fourth power of $\Delta(x)$. The exponents that we obtain in the error terms, namely $\beta = {\sfrac{7}{5}}$ and $\gamma = {\sfrac{23}{12}}$, respectively, are new. They improve on the values $\beta = {\sfrac{47}{28}}, \gamma = {\sfrac{45}{23}}$, due to K.-M. Tsang. A result on integrals of $\Delta^3(x)$ and $\Delta^4(x)$ in short intervals is also proved.

Moments over short intervals

An asymptotic result for the kth moment (k ≦ 9) of the error term in the Dirichlet divisor problem over short intervals is obtained, which improves on an earlier result of Nowak.

On the average orders of the error term in the Dirichlet divisor problem

Let Δ ( x ) be the error term in the Dirichlet divisor problem. The purpose of this paper is to study the difference between two kinds of mean value formulas of Δ ( x ) , that is, the mean value formulas ∫ 1 x Δ ( u ) k du and ∑ n ⩽ x Δ ( n ) k with a natural number k. In particular we study the case k = 2 and 3 in detail.

Exact Asymptotics in log log Laws for Random Fields

Let $$\{ X,X_k ,k \in {\mathbb{N}}^r \}$$ be i.i.d. random variables, and set S n =∑ k ≤ n X k . We exhibit a method able to provide exact loglog rates. The typical result is that $${\mathop {\lim }\limits_{\varepsilon \searrow \sigma \sqrt {2r}} } \sqrt {\varepsilon ^2 - 2r\sigma ^2 } \sum\limits_n {\frac{1}{{|\,n\,|}}P(|S_n \geqslant \varepsilon \sqrt {|\,n\,|\log \log |\,n\,|} ) = \frac{{\sigma \sqrt {2r} }}{{r!}},}$$ whenever EX=0,EX 2=σ2 and E[X 2(log+ | X |) r-1] < ∞. To get this and other related precise asymptotics, we derive some general estimates concerning the Dirichlet divisor problem, of interest in their own right.

Estimates of convolutions of certain number‐theoretic error terms

Several estimates for the convolution function and its iterates are obtained when f(x) is a suitable number‐theoretic error term. We deal with the case of the asymptotic formula for (k = 1, 2), the general Dirichlet divisor problem, the problem of nonisomorphic Abelian groups of given order, and the Rankin‐Selberg convolution.

On higher-power moments of Δ(x) (II)

Let $\Delta(x)$ be the error term of the Dirichlet divisor problem. The asymptotic formula of the integral $\int_1^T\Delta^k(x)dx$ is established for any integer $3\leq k\leq 9$ by an unified method. Similar results are also established for some other well-known error terms in the analytic number theory.

Exponential Sums and Lattice Points III

The Gauss circle problem and the Dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve. In the Bombieri?Iwaniec?Mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice. These coincidences are parametrised by integer points close to certain plane curves, the resonance curves. This paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy, the number of integer points minus the area, as in the latest work on single exponential sums. The Bombieri?Iwaniec?Mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range, and it can in principle be used iteratively. We use a bound obtained by Swinnerton-Dyer's approximation determinant method. In the discrepancy estimate $O(R^K (\log R)^{\Lambda })$ in terms of the maximum radius of curvature $R$, we reduce $K$ from 2/3 (classical) and 46/73 (paper II in this series) to 131/208. The corresponding exponent in the Dirichlet divisor problem becomes $K/2 = 131/416$.

On the divisor problem: Moments of Δ(x) over short intervals

Let as usual ∆(x) denote the error term in the Dirichlet divisor problem, i.e., ∑ n≤x d(n) = x log x+ (2γ − 1)x+∆(x), x a large real variable and γ the Euler-Mascheroni constant. While asymptotics for the square-mean of ∆(x) are classic, K.-M. Tsang [Proc. London 1992] recently established results on the third and fourth power moments. We present short interval variants of such asymptotics. THEOREM 1. For T a large real variable, suppose that T 7→ Λ = Λ(T ) increases with T , satisfies 0 0, lim T→∞ T 1/2+ 0 Λ(T ) = 0. Then, as T →∞, ∫ T+Λ T−Λ ( ∆(t) )3 dt ∼ C3(λ)ΛT , and ∫ T+Λ T−Λ ( ∆(t) )4 dt ∼ C4(λ)ΛT , with explicite positive coefficients C3(λ), C4(λ).

On a generalized divisor problem I

We give a discussion on the properties of Δa(x) (− 1 &lt; a &lt; 0), which is a generalization of the error term Δ(x) in the Dirichlet divisor problem. In particular, we study its oscillatory nature and investigate the gaps between its sign-changes for −½ ≤ a &lt; 0.

On the tails of the limiting distribution function of the error term in the Dirichlet divisor problem

On the tails of the limiting distribution function of the error term in the Dirichlet divisor problem

On the Relationship between the Multidimensional Dirichlet Divisor Problem and the Boundary of Zeros of ζ(s)F

where k is a positive integer, x ≥ 2, τk(n) is the number of solutions in positive integers x1 , . . . , xk of the equation x1 · · · xk = n , ζ(s) is the Riemann zeta-function, s = σ + it , i = −1, and σ = Re s , t = Im s . The multidimensional Dirichlet divisor problem consists in finding the correct order of growth of |Rk(x)| as x → +∞ for any k ≥ 2. Dirichlet posed this problem and proved that |Rk(x)| ≤ x1−1/k(c log x) , (1)

On the error term in the mean square formula for the Riemann zeta-function in the critical strip

For 1/2<σ<1 fixed, letEσ(T) denote the error term in the asymptotic formula for\(\int_0^T {|\zeta (\sigma + it)|^2 dt} \). We obtain some new bounds forEσ(T), and an Ω_-result which is the analogue of the strongest Ω_-result in the classical Dirichlet divisor problem.

The mean square of the error term in a generalization of Dirichlet's divisor problem

The mean square of the error term in a generalization of Dirichlet's divisor problem

Mean square of the remainder term in the Dirichlet divisor problem II

Mean square of the remainder term in the Dirichlet divisor problem II

Mean square of the remainder term in the Dirichlet divisor problem

Mean square of the remainder term in the Dirichlet divisor problem