The triple, quintuple and setuple product identities revisited.
The two parameter ellipse problem
The two parameter hyperbola problem
The value of additive forms at prime arguments
Let be an additive form of degree with prime variables . Suppose that has real coefficients with at least one ratio algebraic and irrational. If s is large enough then takes values close to almost all members of any well-spaced sequence. This complements earlier work of Brüdern, Cook and Perelli (linear forms) and Cook and Fox (quadratic forms). The result is based on Hua’s Lemma and, for , Heath-Brown’s improvement on Hua’s Lemma.
The values of a quadratic form at square-free points
The Waring-Goldbach problem for cubes
The weight of half-integral weight modular forms with few non-vanishing coefficients mod l
Théorème de Miech sur les nombres presque-premiers dans les fonctions polynômes
Theoreme de Pfister pour les varietes et anneaux de Witt reduits.
Théorèmes généraux sur l'impossibilité des équations cubiques indéterminées
Théorie additive des nombres et problème de Waring
Théorie additive et géométrie des nombres
Theorie der Gitterpunkte.
Three additive cubic equations
Three aspects of partitions.
Three restricted product-sum partition functions
Three two-dimensional Weyl steps in the circle problem I. The Hessian determinant
1. Summary. In a sequence of three papers we study the circle problem and its generalization involving the logarithmic mean. Most of the deeper results in this area depend on estimates of exponential sums. For the circle problem itself Chen has carried out such estimates using three two-dimensional Weyl steps with complicated techniques. We make the same Weyl steps but our approach is simpler and clearer. Crucial is a good understanding of the Hessian determinant that appears and a simple...
Three two-dimensional Weyl steps in the circle problem II. The logarithmic Riesz mean for a class of arithmetic functions
1. Summary. In Part II we study arithmetic functions whose Dirichlet series satisfy a rather general type of functional equation. For the logarithmic Riesz mean of these functions we give a representation involving finite trigonometric sums. An essential tool here is the saddle point method. Estimation of the exponential sums in the special case of the circle problem will be the topic of Part III.
Two congruences involving 4-cores.
Two problems associated with convex finite type domains.
We use scaling properties of convex surfaces of finite line type to derive new estimates for two problems arising in harmonic analysis. For Riesz means associated to such surfaces we obtain sharp Lp estimates for p > 4, generalizing the Carleson-Sjölin theorem. Moreover we obtain estimates for the remainder term in the lattice point problem associated to convex bodies; these estimates are sharp in some instances involving sufficiently flat boundaries.