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 $B\left(x\right)$ of primitive lattice points (integer points $(m,n)$ with $gcd(M,n)=1$) in the ellipse disc $Q(u,v)\le x$, in particular, for the remainder term $R\left(x\right)$ in the asymptotics for $B\left(x\right)$. While upper bounds for $R\left(x\right)$ 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...

Let $\varphi (x_1,...,x_h,y)=u_1{x}_1+\cdots +u_h{x}_h+vy$ 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 $\varphi$ and the sets $𝒜$ and $B$ as follows :$$R_{𝒜,B}^{\left(\varphi \right)}\left(n\right)=\text{card}\left(\begin{array}{cc}\hfill \{(a_1,...,a_h,b)\in A_1& \times \cdots \times A_h\times B:\hfill \\ & \varphi (a_1,...,a_h,b)=n\}\hfill \end{array}\right).$$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 $\{\varphi (a_1,...,a_h,0):(a_1,...,a_h)\in A_1\times \cdots \times A_h\}.$ Other results for complementing sets with respect to linear forms are also proved.

Nous montrons dans la première partie l’existence d’un prolongement méromorphe à tout le plan complexe$ℂ$ et explicitons les propriétés et quelques conséquences, d’une large classe de séries zêta des hauteurs associées à l’espace projectif ${\mathbb{P}}_n\left(\mathbb{Q}\right)$$(n\ge 1...$

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...