Positivity of character sums and random multiplicative functions

Quadratic Dirichlet characters play a special role in analytic number theory, because distribution of zeros of their $L$-functions turns out to be connected with general questions on distribution of primes in arithmetic progressions. Let $p$ be a prime number and $\chi_p(\cdot)$ be the corresponding quadratic character $\mod p$, i.e. the Legendre symbol. We will discuss some properties of the set $\mathcal{L}^{+}$ of primes $p$ such that for all positive integers $N$ we have
$$ \chi_p(1)+\ldots+\chi_p(N) \geqslant 0 $$
and present a proof of the estimate
$$ |\mathcal L^+\cap [1,x]|\ll \pi(x)(\ln\ln x)^{-c+o(1)}\text{, where } $$
where
$$ c=2+\sqrt{2}-\frac{\sqrt{23+16\sqrt{2}}}{2}\approx 0.0368, $$
which relies on results of A. Harper on random multiplicative functions.