The distribution of large quadratic character sums and applications (Q6622680)

For any Dirichlet character \(\chi\) modulo \(q\), let \(m(\chi)\) be the normalized maximum character sum, defined by\N\[\Nm(\chi):= \frac{e^{-\gamma} \pi}{\sqrt{q}}\max_{t\leq q} \Big|\sum_{n\leq t} \chi(n)\Big|.\N\]\NLet \(\mathcal{F}(x)\) be the set of fundamental discriminants \(d\) such that \(|d|\leq x\), and let \(\mathcal{F}^{+}(x)\) (respectively \(\mathcal{F}^{-}(x)\)) be the subset of \(\mathcal{F}(x)\) consisting of positive (respectively negative) discriminants. In the paper under review, the author estimates the distribution functions\N\[\N\Psi^{\pm}_x(\tau):=\frac{1}{|\mathcal{F}^{\pm}(x)|}\left|\{d\in \mathcal{F}^{\pm}(x) : m(\chi_d)>\tau\}\right|.\N\]\NMore precisely, he shows that for \(x\) sufficiently large real number, uniformly for \(\tau\) in the range \(2\leq \tau \leq \log_2 x+\log_5 x-\log_4 x- C\) (where \(C>0\) is a suitably large constant) one has\N\[\N\exp\left(-\frac{e^{\alpha}}{ \tau} \left(1+O\left(\frac{\log^2 \tau}{\sqrt{\tau}}\right)\right)\right)\leq \Psi^{-}_x(\tau)\leq \exp\left(-\frac{e^{\beta}}{ \tau} \left(1+O\left(\frac{\log \tau }{\tau}\right)\right)\right),\N\]\Nwhere \(\alpha=\tau-\eta-B_0\) and \(\beta=\tau-\eta -\log 2 - 2\) with \(\eta= e^{-\gamma}\log 2\) and\N\[\NB_0= \int_0^1 \frac{\tanh y}{y} dy + \int_1^{\infty} \frac{\tanh y-1}{y}dy\approxeq 0.8187.\N\]\NRegarding \(\Psi^+_x(\tau)\) he shows that there exists positive constants \(C_1\) and \(C_2\) such that uniformly for \(\tau\) in the range \(2\leq \tau \leq (\log_2 x+\log_5 x-\log_4 x- C_1)/\sqrt{3}\) we have\N\[\N\exp\left(-\frac{e^{\sqrt{3}\tau-B_0}}{\sqrt{3}\tau} \left(1+O\left(\frac{1 }{\tau}\right)\right)\right)\leq \Psi^+_x(\tau)\ll \exp\left(-\frac{e^{\sqrt{3}\tau}}{ \tau^{C_2}}\right).\N\]\NThe author's approach allows him to establish similar approximations concerning \(\Psi_{x, a}^{\text{ prime }}(\tau)\) for \(a\in \{1, 3\}\), where \(\Psi_{x, a}^{\text{ prime }}(\tau)\) is the proportion of primes \(p\leq x\) such that \(p\equiv a \bmod 4\) and \(m((\frac{\cdot}{p}))>\tau\), with \((\frac{\cdot}{p})\) denoting the Legendre symbol. The author also obtains approximations for \(F_{x, a}(\tau)\), which is defined for \(\tau>0\) and \(a\in \{1, 3\}\) by\N\[\NF_{x, a}(\tau):= \frac{2}{\pi(x)} \left|\left\{p\leq x : p\equiv a \bmod 4, \ L\left(1, \left(\frac{\cdot}{p}\right)\right)>e^{\gamma} \tau\right\} \right|.\N\]\NAs an application, he considers a question of Montgomery, approximating the number of primes \(p\leq x\) with \(p\equiv 3\bmod 4\) and such that \(\lambda(p)>1-1/T\), as more as the number of primes \(p\leq x\) with \(p\equiv 3\bmod 4\) and such that \(\lambda(p)<1/3+1/T\), where \(\lambda(p)\) is the Lebesgue measure of the set\N\[\N\left\{\alpha \in [0, 1) : \sum_{0\leq n\leq \alpha p} \left(\frac{n}{p}\right)>0\right\}.\N\]