On the Representations of $xy+yz+zx$

Borwein, Jonathan M. and Choi, Stephen (2000) On the Representations of $xy+yz+zx$. Experimental Mathematics, 9 . pp. 153-158.

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      Recently, Crandall in \cite{Cr} used Andrews' identity for the cube of the Jacobian theta function $\theta_4$: \[\theta_4^3(q)=\left(\sum_{n \in \IZ}(-1)^nq^{n^2}\right)^3=1+4\sum_{n=1}^{\infty}\frac{(-1)^nq^n}{1+q^n}-2\sum_{\substack{n=1 \\ |j| < n}}^{\infty}\frac{q^{n^2-j^2}(1-q^n)(-1)^j}{1+q^n}\] to derive new representations for Madelung's constant and various of its analytic relatives. He considered the three-dimensional Epstein zeta function $M(s)$ which is the analytic continuation of the series \[\sum_{\substack{x,y,z\in \IZ \\ (x,y,z)\neq (0,0,0)}}\frac{(-1)^{x+y+z}}{(x^2+y^2+z^2)^{s}}.\] Then the number $M(\frac{1}{2})$ is the celebrated {\em Madelung constant}. Using a reworking of the above mentioned Andrews' identity, he obtained the formula \[M(s)=-6(1-2^{1-s})^2 \zeta^2(s)-4U(s)\] where $\zeta (s)$ is the Riemann zeta function and \[U(s):=\sum_{x,y,z \ge 1}\frac{(-1)^{x+y+z}}{(xy+yz+xz)^s}.\] In view of this representation, Crandall asked what integers are of the form of $xy+yz+xz$ with $x,y,z \ge 1$ and he made a conjecture that every odd integer greater than one can be written as $xy+yz+xz$. In this manuscript, we shall show that Crandall's conjecture is indeed true.

      Item Type: Article
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      Date Deposited: 25 Nov 2003
      Last Modified: 13 Jan 2015 12:41

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