A propos du problème de Waring
A quantitative aspect of non-unique factorizations: the Narkiewicz constants II
Let K be an algebraic number field with non-trivial class group G and be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let denote the number of non-zero principal ideals with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that behaves, for x → ∞, asymptotically like . In this article, it is proved that for every prime p, , and it is also proved that if and m is large enough. In particular, it is shown that for...
A Rademacher type formula for partitions and overpartitions.
A reduction technique in Waring's problem, I
A refinement of a partition theorem of Sellers.
A remark on a previous paper by Bredihin and Linnik
A remark on Chen's theorem
A remark on the Piatetski-Shapiro-Vinogradov theorem
A remark on the quadratic analogue of the Quillen-Suslin theorem.
A set of squares without arithmetic progressions
A sharp estimate of the number of integral points in a 4-dimensional tetrahedra.
A sharp estimate of the number of integral points in a tetrahedron.
A short intervals result for linear equations in two prime variables.
Given A and B integers relatively prime, we prove that almost all integers n in an interval of the form [N, N+H], where N exp(1/3+e) ≤ H ≤ N can be written as a sum Ap1 + Bp2 = n, with p1 and p2 primes and e an arbitrary positive constant. This generalizes the results of Perelli et al. (1985) established in the classical case A=B=1 (Goldbach's problem).
A sieve approach to the Waring-Goldbach problem. I. Sums of four cubes
A sieve approach to the Waring-Goldbach problem, II On the seven cubes theorem
A simple proof of the Ramanujan conjecture for powers of 5.
A singular series average and Goldbach numbers in short intervals
A special case of Vinogradov's mean value theorem
A structure theorem for sets of small popular doubling
We prove that every set A ⊂ ℤ satisfying for t and δ in suitable ranges must be very close to an arithmetic progression. We use this result to improve the estimates of Green and Morris for the probability that a random subset A ⊂ ℕ satisfies |ℕ∖(A+A)| ≥ k; specifically, we show that .
A structure theory for small sum subsets