Real zeros of Dedekind zeta functions of real quadratic fields

Abstract:

Let $\chi$ be a primitive, real and even Dirichlet character with conductor $q$, and let $s$ be a positive real number. An old result of H. Davenport is that the cycle sums $S_\nu (s,\chi )=\sum _{n=\nu q+1}^{(\nu +1)q-1} \frac {\chi (n)}{n^s}, \nu = 0,1,2,\dots ,$ are all positive at $s=1,$ and this has the immediate important consequence of the positivity of $L(1,\chi )$. We extend Davenport’s idea to show that in fact for $\nu \geq 1$, $S_\nu (s,\chi )>0$ for all $s$ with $1/2 \leq s \leq 1$ so that one can deduce the positivity of $L(s,\chi )$ by the nonnegativity of a finite sum $\sum _{\nu =0}^t S_\nu (s,\chi )$ for any $t \geq 0$. A simple algorithm then allows us to prove numerically that $L(s,\chi )$ has no positive real zero for a conductor $q$ up to 200,000, extending the previous record of 986 due to Rosser more than 50 years ago. We also derive various estimates explicit in $q$ of the $S_\nu (s,\chi )$ as well as the shifted cycle sums $T_\nu (s,\chi ):=\sum _{n=\nu q+\lfloor q/2 \rfloor +1}^{(\nu +1) q+\lfloor q/2 \rfloor } \frac {\chi (n)}{n^s}$ considered previously by Leu and Li for $s=1$. These explicit estimates are all rather tight and may have independent interests.