Schur's determinants and partition theorems.
Self-conjugate vector partitions and the parity of the spt-function
Let spt(n) denote the total number of appearances of the smallest parts in all the partitions of n. Recently, we found new combinatorial interpretations of congruences for the spt-function modulo 5 and 7. These interpretations were in terms of a restricted set of weighted vector partitions which we call S-partitions. We prove that the number of self-conjugate S-partitions, counted with a certain weight, is related to the coefficients of a certain mock theta function studied by the first author,...
Semisemiological structure of the prime numbers and conditional Goldbach theorems.
Septic analogues of the Rogers-Ramanujan functions
Series and iterations for 1/π
Sets of parts such that the partition function is even
Sets with even partition functions and 2-adic integers. II.
Simultaneous quadratic equations II
Simultaneous quadratic inequalities
Singularité de séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres
Soit un polynôme. On appelle série de Dirichlet associée à la fonction : . Dans cet article nous étudions l’existence et les propriétés du prolongement méromorphe d’une telle série sous l’hypothèse qu’il existe tel que : i) quand et et ii) où . Cette hypothèse est probablement optimale et en tout cas contient strictement toutes les classes de polynômes déjà traitées antérieurement. Sous cette hypothèse nos principaux résultats sont : l’existence du prolongement méromorphe au plan...
Slim exceptional sets for sums of fourth and fifth powers
Small prime solutions of linear ternary equations
Small Prime Solutions of some Additive Equations.
Small prime solutions of ternary linear equations
Small solutions of cubic equations with prime variables in arithmetic progressions
Small solutions to linear congruences and Hecke equidistribution
Solvability of 𝔭-adic diagonal equations
Solving a ± b = 2c in elements of finite sets
We show that if A and B are finite sets of real numbers, then the number of triples (a,b,c) ∈ A × B × (A ∪ B) with a + b = 2c is at most (0.15+o(1))(|A|+|B|)² as |A| + |B| → ∞. As a corollary, if A is antisymmetric (that is, A ∩ (-A) = ∅), then there are at most (0.3+o(1))|A|² triples (a,b,c) with a,b,c ∈ A and a - b = 2c. In the general case where A is not necessarily antisymmetric, we show that the number of triples (a,b,c) with a,b,c ∈ A and a - b = 2c is at most (0.5+o(1))|A|². These estimates...
Some additive problems of numbers
Some arithmetic properties of overpartition -tuples.