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Self-conjugate vector partitions and the parity of the spt-function

George E. Andrews, Frank G. Garvan, Jie Liang (2013)

Acta Arithmetica

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,...

Singularité de séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres

Driss Essouabri (1997)

Annales de l'institut Fourier

Soit P ∈ ℝ [ X 1 , ... , X n ] un polynôme. On appelle série de Dirichlet associée à P la fonction : s ↦ Z ( P ; s ) = ∑ m ∈ ℕ * n P ( m ) - s ( s ∈ ℂ ) . Dans cet article nous étudions l’existence et les propriétés du prolongement méromorphe d’une telle série sous l’hypothèse qu’il existe B ∈ ] 0 , 1 [ tel que : i) P ( x ) → + ∞ quand | | x | | → + ∞ et x ∈ [ B , + ∞ [ n et ii) d ( Z ( P ) , [ B , + ∞ [ n ) > 0 où Z ( P ) = { z ∈ ℂ n | P ( z ) = 0 } . 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...

Solving a ± b = 2c in elements of finite sets

Vsevolod F. Lev, Rom Pinchasi (2014)

Acta Arithmetica

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...

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