Low-lying zeros of dihedral \(L\)-functions (Q1872812)

Let \(D>3\) be squarefree, \(D \equiv 3 \pmod{4}\), \(\psi \) a character of the ideal class group of the imaginary quadratic field \(\mathbb Q (\sqrt{-D})\), and consider the zeros near \(s=1/2\) of the \(L\)-functions \(L(s,\psi),\) assuming the Riemann hypothesis for such functions. There is a philosophy, due to N. Katz and P. Sarnak, predicting the distribution of low-lying zeros of various families of \(L\)-functions, and accordingly a ``Density Conjecture'' is formulated in the present case. This conjecture involves a test function of compact support, and the goal is to prove it for test functions with support as wide as possible. Results in this direction are obtained, summing either over all character for fixed \(D\), or averaging even over \(D\). Arithmetically, the Density Conjecture appears to be equivalent to an ``Euler Primes Conjecture'' concerning primes \(p\) such that \(4p = m^2+Dn^2\). Sieve methods are applied to count such primes.