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We call a quadratic character $\chi$ positive definite if the partial sum $\sum_{1\leq n\le N}\chi(n)$ is nonnegative for all $N$. If we enumerate all quadratic characters by its conductor $d\leq D$, it is known that the density of positive definite characters is asymptotically zero, due to a result of Baker and Montgomery (1989). We gave a quantitative improvements, showing that the density is 

at most $(\log D)^{-c}$ for a positive constant $c$ and we believe the bound is sharp up to the constant c. This is joint work in progress with R. Angelo and K. Soundararajan.