Changes of sign of error terms related to Euler's function and to divisor functions.
Circles passing through five or more integer points
We find an improvement to Huxley and Konyagin’s current lower bound for the number of circles passing through five integer points. We conjecture that the improved lower bound is the asymptotic formula for the number of circles passing through five integer points. We generalise the result to circles passing through more than five integer points, giving the main theorem in terms of cyclic polygons with m integer point vertices. Theorem. Let m ≥ 4 be a fixed integer. Let be the number of cyclic polygons...
Comment distribuer des points uniformément sur une sphère ? (D'après Lubotzky, Phillips et Sarnak)
Continuity Properties of Voronoi Domains.
Corrigendum to “On short sums of certain multiplicative functions”.
Counting problems and semisimple groups.
Crible et 3-rang des corps quadratiques
Considérons le cardinal de l’ensemble des racines cubiques de l’unité dans le groupe des classes de , où est un discriminant fondamental. Un résultat de Davenport et Heilbronn calcule la valeur moyenne de ces nombres quand varie. On obtient ici géométriquement une borne explicite pour le reste, avec la possibilité supplémentaire de restreindre les à des progressions arithmétiques. Des techniques de crible permettent alors d’évaluer la 3-partie des , où est pseudo-premier d’ordre . On...
Critical Dimensions for counting Lattice Points in Euclidean Annuli
We study the number of lattice points in ℝd, d ≥ 2, lying inside an annulus as a function of the centre of the annulus. The average number of lattice points there equals the volume of the annulus, and we study the L1 and L2 norms of the remainder. We say that a dimension is critical, if these norms do not have upper and lower bounds of the same order as the radius goes to infinity. In [Duke Math. J., 107 (2001), No. 2, 209–238], it was proved that...
Croissance de certaines séries de Dirichlet et applications.
Cyclic polygons of integer points