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Primitive lattice points inside an ellipse

Werner Georg Nowak (2005)

Czechoslovak Mathematical Journal

Let Q ( u , v ) be a positive definite binary quadratic form with arbitrary real coefficients. For large real x , one may ask for the number B ( x ) of primitive lattice points (integer points ( m , n ) with gcd ( M , n ) = 1 ) in the ellipse disc Q ( u , v ) x , in particular, for the remainder term R ( x ) in the asymptotics for B ( x ) . While upper bounds for R ( x ) depend on zero-free regions of the zeta-function, and thus, in most published results, on the Riemann Hypothesis, the present paper deals with a lower estimate. It is proved that the absolute value or...

Problems in additive number theory, II: Linear forms and complementing sets

Melvyn B. Nathanson (2009)

Journal de Théorie des Nombres de Bordeaux

Let ϕ ( x 1 , ... , x h , y ) = u 1 x 1 + + u h x h + v y be a linear form with nonzero integer coefficients u 1 , ... , u h , v . Let 𝒜 = ( A 1 , ... , A h ) be an h -tuple of finite sets of integers and let B be an infinite set of integers. Define the representation function associated to the form ϕ and the sets 𝒜 and B as follows : R 𝒜 , B ( ϕ ) ( n ) = card { ( a 1 , ... , a h , b ) A 1 × × A h × B : ϕ ( a 1 , ... , a h , b ) = n } . If this representation function is constant, then the set B is periodic and the period of B will be bounded in terms of the diameter of the finite set { ϕ ( a 1 , ... , a h , 0 ) : ( a 1 , ... , a h ) A 1 × × A h } . Other results for complementing sets with respect to linear forms are also proved.

Proof of a conjecture of Hirschhorn and Sellers on overpartitions

William Y. C. Chen, Ernest X. W. Xia (2014)

Acta Arithmetica

Let p̅(n) denote the number of overpartitions of n. It was conjectured by Hirschhorn and Sellers that p̅(40n+35) ≡ 0 (mod 40) for n ≥ 0. Employing 2-dissection formulas of theta functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating function for p̅(40n+35) modulo 5. Using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we prove the congruence p̅(40n+35) ≡ 0 (mod 5) for n ≥ 0. Combining this congruence and the congruence p̅(4n+3) ≡ 0 (mod...

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