On the number of primitive Pythagorean triangles
Improving on some results of J.-L. Nicolas [15], the elements of the set , for which the partition function (i.e. the number of partitions of with parts in ) is even for all are determined. An asymptotic estimate to the counting function of this set is also given.
Let λ denote Carmichael’s function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler’s φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds for all large x, while for φ it is equal to , an old result of Erdős. We also improve on an earlier result of the first author and Friedlander giving an upper bound for the distribution of λ-values.
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 . We also show how our method of proof yields the bound , where T 1/5+ε≤G≪T, T
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 and It is also shown how bounds for moments of | E *(t)| lead to bounds for moments of .
Let ϕ(n) denote the Euler totient function. We study the error term of the general kth Riesz mean of the arithmetical function n/ϕ(n) for any positive integer k ≥ 1, namely the error term where . For instance, the upper bound for |Ek(x)| established here improves the earlier known upper bounds for all integers k satisfying .
One of the oldest problems in analytic number theory consists of counting points with integer coordinates in the d-dimensional ball. It is very easy to find a main term for the counting function, but the size of the error term is difficult to estimate (...).
Let F be a Siegel cusp form of integral weight k on the Siegel modular group Sp₂(ℤ) of genus 2. The coefficients of the spinor zeta function are denoted by cₙ. Let be the Riesz mean of cₙ. Kohnen and Wang obtained the truncated Voronoï-type formula for under the Ramanujan-Petersson conjecture. In this paper, we study the higher power moments of , and then derive an asymptotic formula for the hth (h=3,4,5) power moments of by using Ivić’s large value arguments and other techniques.