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Let $k$ be a fixed integer. We study the asymptotic formula of $R (H, r, k)$ , which is the number of positive integer solutions $1 \leq x, y, z \leq H$ such that the polynomial $x^{2} + y^{2} + z^{2} + k$ is $r$ -free. We obtained the asymptotic formula of $R (H, r, k)$ for all $r \geq 2$ . Our result is new even in the case $r = 2$ . We proved that $R (H, 2, k) = c_{k} H^{3} + O (H^{9 / 4 + ε})$ , where $c_{k} > 0$ is a constant depending on $k$ . This improves upon the error term $O (H^{7 / 3 + ε})$ obtained by G.-L. Zhou, Y. Ding (2022).

On the sphere problem.

Fernando Chamizo, Henryk Iwaniec (1995)

Revista Matemática Iberoamericana

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 (...).

In this paper, we are interested in exploring the cancellation of Hecke eigenvalues twisted with an exponential sums whose amplitude is √n at prime arguments.

Displaying 41 – 51 of 51

On the number of Abelian groups of a given number.

On the numbers that are representable as the sum of two cubes.

On the order of $ζ (\frac{1}{2} + i t)$

On the $p$ -adic continuation of the logarithmic derivative of certain hypergeometric functions

On the $r$ -free values of the polynomial $x^{2} + y^{2} + z^{2} + k$

On the sphere problem.

On the sums $S_{χ} (m)$

On the zeta function of monodromy of a polynomial map

On Zeros of Dirirchlet's L Series.

Optimal bounds for exponential sums in terms of discrepancy

Oscillations of Hecke eigenvalues at shifted primes.

Currently displaying 41 – 51 of 51