Abstract:
Using the second moment of Dirichlet $L$-functions on the critical line over the major arcs ${\mathfrak M}({\mathscr L}^b)$, with $\tau=y^3x^{-1}{\mathscr L}^{-b_1}$, and excluding a small neighborhood of the centers of these arcs, i.e., those $\alpha$ satisfying $|\alpha-\frac{a}{q}|>(8\pi y^2)^{-1}$, for $y\ge x^{1-\frac{\scriptstyle1}{\scriptstyle{9-4\sqrt{2}}}}{\mathscr L}^{c_2}$, where $c_2=\frac{2A+24+(\sqrt{2}-1)b_1}{2\sqrt{2}-1}$, we obtain the estimate $$ S_2(\alpha;x,y)=\sum_{x-y<n\le x}\Lambda(n)e(\alpha n^2)\ll y {\mathscr L}^{-A}. $$ Moreover, in a small neighborhood of the center of the major arcs, defined by $|\alpha-\frac{a}{q}|\le(8\pi y^2)^{-1}$, for $y\ge x^{\frac{\scriptstyle5}{\scriptstyle8}}{\mathscr L}^{1,5A+0,25b+18}$, an asymptotic formula with a remainder term is obtained for $S_2(\alpha;x,y)$, where $A$, $b_1$, and $b$ are arbitrary fixed positive constants, and ${\mathscr L}=\ln x$.
Keywords:Short exponential sum with primes, major arcs, density theorem, Dirichlet $L$-function.
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