二次特征 theta 函数的对数凹性:无穷幂增长族与极值情形(d=5) Log-concavity of theta functions of quadratic characters: an infinite family of power growth and the extremal case (d=5)

For primes \(d\equiv1\pmod 4\) let \(\theta_{\chi_d}(u)=\sum_{n\ge1}\chi_d(n)e^{-\pi n^2u/d}\) and \(L_d(t)=\log\theta_{\chi_d}(e^t)\). We study the set \(\E\) of those \(d\) for which \(\theta_{\chi_d}\) is log-concave on \((0,\infty)\), equivalently \(L_d''\le0\). We prove two elementary universal facts. First, \(L_d'(0)=-\tfrac14\) for every \(d\). Second, \(L_d''(\log d)=-\pi+3\pi(3\pi-1)\chi_d(2)e^{-3\pi}+O(10^{-6})\), and consequently \(L_d''(t)\le-3.13\) for every \(t\ge\log d\). Together with the functional equation these reduce the verification of log-concavity to the finite window \(t\in[0,\log d)\). We also prove that \(L_d'''<0\) on the whole half-axis \(t\ge\log d\), so that the functional \(F=tL_d''-L_d'\) is strictly decreasing there; for \(d=5\) this combines with an interval certificate on the finite window to give \(F<\tfrac14\) for all \(t>0\). Numerically, among the \(39\,175\) primes \(d\equiv1\pmod4\) with \(d\le10^6\), our computation finds \(777\) members of \(\E\), and the counting function grows like \(X^{0.616}\) over three decades; we conjecture \(\#\E(X)\asymp X^{\alpha}\) with \(\alpha\approx0.62\), in particular that \(\E\) is infinite. The largest concavity margin occurs at \(d=5\), where \(\max_tL_5''=-1.2225\), that is \(3.385\) times the second largest value (at \(d=17\)). Failures split into three numerically stable classes. Since certified inclusions are presently out of reach, the counts are upper bounds; we state precisely which assertions are proved, which are certified by interval arithmetic, and which are numerical.