On two classical lattice point problems

1940 ◽  
Vol 36 (2) ◽  
pp. 131-138 ◽  
Author(s):  
A. E. Ingham
Keyword(s):  
Lattice Point ◽  
Error Term ◽  
Divisor Problem ◽  
Lattice Points ◽  
Classical Lattice ◽  
Rectangular Hyperbola ◽  
Number Of Divisors ◽  
Image Position ◽  
Euler's Constant ◽  
Lattice Point Problems

1. Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Thus P(x) is the error term in the problem of the lattice points of a circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of a rectangular hyperbola.

Two classical lattice point problems

1961 ◽  
Vol 57 (4) ◽  
pp. 699-721 ◽  
Author(s):  
K. S. Gangadharan ◽  
A. E. Ingham
Keyword(s):  
Lattice Point ◽  
Error Term ◽  
Divisor Problem ◽  
Lattice Points ◽  
Classical Lattice ◽  
Rectangular Hyperbola ◽  
Number Of Divisors ◽  
Image Position ◽  
Euler's Constant ◽  
Lattice Point Problems

Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Then P(x) is the error term in the problem of the lattice points of the circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of the rectangular hyperbola.

scholarly journals On Dirichlet's divisor problem

1931 ◽  
Vol 134 (823) ◽  
pp. 192-202 ◽  
Keyword(s):  
Positive Integer ◽  
Divisor Problem ◽  
Euler’S Constant ◽  
Number Of Divisors ◽  
Prime Factors ◽  
Euler's Constant

1. Let d ( n ) denote the number of divisors of the positive integer n , so that, if n = p 1 a 1 . . . p r ar is the canonical expression of n in prime factors, d ( n ) = (1 + a 1 ) . . . (1 + a r ), and let d ( x ) = 0 if x is not an iteger; then if (1. 1) D ( x ) = Σ' n ≤ x d ( n ) = Σ n ≤ x d ( n ) ─ ½ d ( x ), and (1. 2) Δ ( x ) = D ( x ) ─ x log x ─ (2C ─ 1) x ─ ¼, where C is Euler's constant, it was proved by Dirichlet in 1849 that (1. 21) Δ ( x ) = O (√ x ),

Cushions, cigars and diamonds: an area-perimeter problem for symmetric ovals

1979 ◽  
Vol 85 (1) ◽  
pp. 1-16 ◽  
Author(s):  
H. T. Croft
Keyword(s):  
Upper Bound ◽  
Natural Condition ◽  
Lattice Point ◽  
Convex Set ◽  
General Setting ◽  
Lattice Points ◽  
Two Dimensional ◽  
Algebraic Expressions ◽  
Image Position ◽  
Closed Convex Set

P. R. Scott (1) has asked which two-dimensional closed convex set E, centro-symmetric in the origin O, and containing no other Cartesian lattice-point in its interior, maximizes the ratio A/P, where A, P are the area, perimeter of E; he conjectured that the answer is the ‘rounded square’ (‘cushion’ in what follows), described below. We shall prove this, indeed in a more general setting, by seeking to maximizewhere κ is a parameter (0 < κ < 2); the set of admissible E is those E centro-symmetric in 0 that do not contain in their interior certain fixed lattice-points. There are two problems, the unrestricted one , where there is no given upper bound on A (it will become apparent that this problem only has a finite answer when κ ≥ 1) and the restricted one , when one is given a bound B and we must have A ≤ B. Special interest attaches to the case B = 4, both because of Minkowski's theorem: any E symmetric in O and containing no other lattice-point has area at most 4; and because it turns out that it is a ‘natural’ condition: the algebraic expressions simplify to a remarkable extent. Hence in what follows, the ‘restricted case ’ shall mean A ≤ 4.

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

2016 ◽  
Vol 12 (08) ◽  
pp. 2231-2239
Author(s):  
Aleksandar Ivić
Keyword(s):  
Zeta Function ◽  
Error Term ◽  
Upper Bounds ◽  
Mean Value ◽  
Divisor Problem ◽  
Asymptotic Formulas ◽  
Number Of Divisors ◽  
The Riemann Zeta Function ◽  
Dirichlet Divisor Problem ◽  
Riemann Zeta

Let [Formula: see text] be the number of divisors of [Formula: see text], let [Formula: see text] denote the error term in the classical Dirichlet divisor problem, and let [Formula: see text] denote the Riemann zeta-function. Several upper bounds for integrals of the type [Formula: see text] are given. This complements the results of [A. Ivić and W. Zhai, On some mean value results for [Formula: see text] and a divisor problem II, Indag. Math. 26(5) (2015) 842–866], where asymptotic formulas for [Formula: see text] were established for the above integral.

On lattice points in the domain |xy | ≤1, | x + y | ≤√5, and applications to asymptotic formulae in lattice point theory (II)

1944 ◽  
Vol 40 (2) ◽  
pp. 116-120 ◽  
Author(s):  
K. Mahler
Keyword(s):  
Lattice Point ◽  
Point Theory ◽  
Lattice Points ◽  
Linear Substitution ◽  
Asymptotic Formulae ◽  
Image Position ◽  
Similar Domain

I. Lattice points in the domain | x |α + | y |α ≤ 1Theorem 4. Let G be the star domainwhere α > 0. Then, when α tends to zero,Proof. The linear substitutionchanges G into the similar domainand so.

On a Result of Smith and Subbarao Concerning a Divisor Problem

1984 ◽  
Vol 27 (4) ◽  
pp. 501-504 ◽  
Author(s):  
Werner Georg Nowak
Keyword(s):  
Positive Integer ◽  
Recent Result ◽  
Divisor Problem ◽  
Number Of Divisors ◽  
Image Position

AbstractLet d(n;l,k) denote the number of divisors of the positive integer n which are congruent to I modulo k. The objective of the present paper is to prove that (for some exponent θ<⅓)holds uniformly in l, k and x satisfying 1≤l≤k≤x. This improves a recent result due to R. A. Smith and M. V. Subbarao [3].

scholarly journals On Measures of Polynomials in Several Variables: Corrigendum

1982 ◽  
Vol 26 (2) ◽  
pp. 317-319 ◽  
Author(s):  
Gerald Myerson ◽  
C.J. Smyth
Keyword(s):  
Error Term ◽  
Euler’S Constant ◽  
Several Variables ◽  
Explicit Constant ◽  
Image Position ◽  
Euler's Constant

In [2], it was asserted in Theorem 3 that the measure M(x0 + … + xx) is asymptotically , where c was an explicit constant. The value of c given was incorrect, and should be e-½γ where γ is Euler's constant. This was pointed out by the first author. In factwhere we have tried to make amends by improving the error term.

scholarly journals Bounds on the Lattice Point Enumerator via Slices and Projections

2021 ◽  
Author(s):  
Ansgar Freyer ◽  
Martin Henk
Keyword(s):  
Convex Body ◽  
Lattice Point ◽  
Geometric Mean ◽  
Discrete Analogue ◽  
Lattice Points ◽  
General Context ◽  
General Bound

AbstractGardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.

Sign in / Sign up

Export Citation Format

Share Document