On the general asymmetric divisor problem

To ProfessorEkkehard Kr?tzel on his 60^th birthday     

Problems similar to the additive divisor problem

For multiplicative functions ?(n), let the following conditions be satisfied: ?(n)≥0 ?(p ^r)≤A ^r,A>0, and for anyε>0 there exist constants $A_\varepsilon$ ,α>0 such that $f(n) \leqslant A_\varepsilon n^\varepsilon$ and Σ _p≤x ?(p) lnp≥αx. For such functions, the following relation is proved: $$\sum\limits_{n \leqslant x} {f(n)} \tau (n - 1) = C(f)\sum\limits_{n \leqslant x} {f(n)lnx(1 + 0(1))}$$ . Hereτ(n) is the number of divisors ofn andC(?) is a constant.     

Problems similar to the additive divisor problem

For multiplicative functions ƒ(n), let the following conditions be satisfied: ƒ(n)≥0 ƒ(p ^r)≤A ^r,A>0, and for anyε>0 there exist constants ,α>0 such that and Σ _p≤x ƒ(p) lnp≥αx. For such functions, the following relation is proved:

The general divisor problem

Let Δ_k(x) = Δ(a₁, …, a_k; x) be the error term in the asymptotic formula for the summatory function of the general divisor function d(a₁, …, a_k; n), which is generated by ζ(a₁s) … ζ(a_ks) (1 ≤ a₁ ≤ … ≤ a_k are fixed integers). Precise omega results for the mean square integral ∫₁^x Δ_k²(x) dx are given, with applications to some specific divisor problems.     

On the ingham divisor problem

The main result of this paper is the following theorem. Suppose thatτ(n) = ∑ _d|n l and the arithmetical functionF satisfies the following conditions:

An additive divisor problem with a growing number of factors

Let τk(n) be the number of representations ofn as the product ofk positive factors, τ(n)=τ(n). The asymptotics of Σ _n≤x τ _k (n)τ(n+1) for 80k ¹⁰ (lnlnx)³≤lnx is shown to be uniform with respect tok. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 391–406, March, 1997. Translated by N. K. Kulman     

Sums of the additive divisor problem type and the inner product method

The inner product approach to the additive divisor problem with its cusp form analogs is surveyed, and a spectral summation formula for convolution sums involving Fourier coefficients of Maass forms is derived. An application to subconvexity estimates for Rankin-Selberg L-functions is announced. Bibliography: 18 titles. Dedicated to the memory of Yu. V. Linnik Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 239–250.     

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 . If with , then we obtain

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.     

On general divisor problems involving Hecke eigenvalues

We study two general divisor problems related to Hecke eigenvalues of classical holomorphic cusp forms, which have been considered by Fomenko, and by Kanemitsu, Sankaranarayanan and Tanigawa respectively. We improve previous results.     

On Riesz means in the multi-dimensional divisor problem

An estimate of the remainder term in an asymptotic formula for Riesz means of the multidimensional divisor function is obtained with an arbitrary value of the parameter α > 0.     

On the riemann zeta-function and the divisor problem II

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 . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain

On the Vinogradov additive problem

Let us state the main result of the paper. Suppose that the collection N ₁, ..., N _n is admissible. Then, in the representation $$ \left\{ \begin{gathered} p_1 + p_2 + \cdots + p_k = N_1 , \hfill \\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \hfill \\ p_1^n + p_2^n + \cdots + p_k^n = N_n , \hfill \\ \end{gathered} \right. $$ where the unknowns p ₁, p ₂, ..., p _k take prime values under the condition p _s > n+ 1, s = 1, ..., k, the number k is of the form $$ k = k_0 + b\left( n \right)s, $$ where s is a nonnegative integer. Further, if k ₀ ≥ a, then, in the representation for k, we can set s = 0, but if k ₀ ≤ a ? 1, then, for a given k ₀ there exist admissible collections (N ₁, ..., N _n ) that cannot be expressed as k ₀ summands of the required form, but can be expressed as k ₀ + b(n) summands.     

The zeta function of the additive divisor problem and spectral decomposition of the automorphic Laplacian

A representation is obtained for the zeta function of the additive divisor problem;, by means of the spectral characteristics of the automorphic Laplacian. On the basis of this representation, the meromorphic continuability of _k(s) to the whole complex plane is proved and a power estimate of the growth of _k(s) as |s| in the critical strip 0es1 is obtained. From this, with the help of the method of complex integration, the asymptotic formula, is derived, where P_k (x) is a quadratic polynomialTranslated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 134, pp. 84–116, 1984.     

On the numerical treatment of a small divisor problem

Summary Quasiperiodic solutions of perturbed integrable Hamiltonian equations such as weakly coupled harmonic oscillators can be found by constructing an appropriate coordinate transformation which leads to a small divisor problem. However the numerical difficulties are not merely caused by the small divisors but rather by the appearence of ghost solutions, which appear in any reasonable discretization of the problem. Our numerical treatment, based on a Newton-type iteration, guarantees an approximation of the relevant solution of the nonlinear problem. Numerical solutions are found up to a critical value of the coupling constant, which is much larger than the coupling constants allowed by the existence theory available so far.