A note on the values of the Epstein zeta functions at the critical points (Q2712562)

For \(K \geq 1\) a fixed natural number let NEWLINE\[NEWLINE Z_K(s) = \mathop{{\sum}'}_{-\infty<m_1,\cdots,m_k<\infty}(m_1^2 + \cdots + m_k^2)^{-s} \qquad(\Re s > K/2), NEWLINE\]NEWLINE where the dash indicates that \((m_1,\cdots,m_k) = (0,\cdots,0)\) is excluded from summation. In view of the functional equation NEWLINE\[NEWLINE \pi^{-s}\Gamma(s)Z_K(s) = \pi^{-(K/2-s)}\Gamma(K/2-s)Z_K(K/2-s) NEWLINE\]NEWLINE it transpires that \(s = K/4\) is the critical point of \(Z_K(s)\). The author's main result is that \(Z_K(K/4) > 0\) for \(K \geq 10\) and \(Z_K(K/4) < 0\) for \(1 \leq K \leq 9\). In fact the author derives explicit closed-form expressions (separate for \(K\) even or odd) for \(Z_K(K/4)\) involving the \(K\)-Bessel function and the arithmetic function NEWLINE\[NEWLINE r_K(n) = \sum_{n=m_1^2+\cdots+m_k^2}1, NEWLINE\]NEWLINE generated by \(Z_K(s)\). A connection between the values of \(Z_K(K/4)\) for \(K\geq 6\) and the values of \(\zeta(2m + 1)\) is noted.