On a certain sum in number theory. III.
On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function
It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function which is the number of positive integers and free of prime factors . Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional () real right-angled simplices. In this paper, we...
On a Problem Connected with a Theorem of Jarnik.
On certain sum in number theory
On Estermann's proof of a theorem of Minkowski
On higher-power moments of Δ(x)
On integer points in polygons
The phenomenon of anomaly small error terms in the lattice point problem is considered in detail in two dimensions. For irrational polygons the errors are expressed in terms of diophantine properties of the side slopes. As a result, for the -dilatation, , of certain classes of irrational polygons the error terms are bounded as with some , or as with arbitrarily small .
On lattice points in large convex bodies
On primitive lattice points in planar domains
On short sums of certain multiplicative functions.
On sums and differences of two coprime kth powers
On Sums of Two Coprime k-th Powers.
On sums of two cubes: an Ω₊-estimate for the error term
The arithmetic function counts the number of ways to write a natural number n as a sum of two kth powers (k ≥ 2 fixed). The investigation of the asymptotic behaviour of the Dirichlet summatory function of leads in a natural way to a certain error term which is known to be in mean-square. In this article it is proved that as t → ∞. Furthermore, it is shown that a similar result would be true for every fixed k > 3 provided that a certain set of algebraic numbers contains a sufficiently...
On sums of two kth powers: a mean-square asymptotics over short intervals
On sums of two kth powers: an asymptotic formula for the mean square of the error term
On the asymmetric divisor problem with congruence conditions
A certain generalized divisor function is studied which counts the number of factorizations of a natural number into integer powers with prescribed exponents under certain congruence restrictions. An -estimate is established for the remainder term in the asymptotic for its Dirichlet summatory function.
On the average order of the lattice rest of a convex body
On the circle problem with general weight
On the circle problem with polynomial weight
On the class number of binary quadratic forms: An omega estimate for the error term.