An additive problem in the theory of numbers
An additive problem with primes and almost-primes
An asymptotic formula for the number of representations in Waring's problem.
An asymptotic formula related to the divisors of the quaternary quadratic form
Let d(n) stand for the Dirichlet divisor function. We give an asymptotic formula for with the help of the circle method.
An ideal Waring problem with restricted summands
An Improved Bound for Prime Solutions of Some Ternary Equations.
An improved upper bound for G(k) in Waring's problem for relatively small k
Application of the circle method on multidimensional limit-periodic functions
Arithmetic progressions in sumsets
1. Introduction. Let A,B ⊂ [1,N] be sets of integers, |A|=|B|=cN. Bourgain [2] proved that A+B always contains an arithmetic progression of length . Our aim is to show that this is not very far from the best possible. Theorem 1. Let ε be a positive number. For every prime p > p₀(ε) there is a symmetric set A of residues mod p such that |A| > (1/2-ε)p and A + A contains no arithmetic progression of length (1.1). A set of residues can be used to get a set of integers in an obvious way. Observe...
Arithmetic progressions of prime-almost-prime twins
Asymptotic formulae for partition ranks
Using an extension of Wright's version of the circle method, we obtain asymptotic formulae for partition ranks similar to formulae for partition cranks which where conjectured by F. Dyson and recently proved by the first author and K. Bringmann.
Average Frobenius distribution of elliptic curves
Bihomogeneous forms in many variables
We count integer points on varieties given by bihomogeneous equations using the Hardy-Littlewood method. The main novelty lies in using the structure of bihomogeneous equations to obtain asymptotics in generically fewer variables than would be necessary in using the standard approach for homogeneous varieties. Also, we consider counting functions where not all the variables have to lie in intervals of the same size, which arises as a natural question in the setting of bihomogeneous varieties.
Bounds for solutions of additive equations in an algebraic number field I
Bounds for solutions of additive equations in an algebraic number field II
Bounds for solutions of two additive equations of odd degree [Book]
Bounds of prime solutions of some diagonal equations.
Conditions suffisantes d'équirépartition modulo 1 . Problème de Waring-Golbach pour f (x) == Xc , c non entier.
Conditions suffisantes d’équirépartition modulo . Problème de Waring-Goldbach pour , non entier
Corrigendum of A Note on the Exceptional Set for Goldbach's Problem in Short Intervals.