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Relative Trace Formula and Twisted $L$-functions: the Burgess Bound
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abstract
Let $F$ be a number field, $\pi$ either a unitary cuspidal automorphic representation of $\mathrm{GL}(2)/F$ or a unitary Eisenstein series, and $\chi$ a unitary Hecke character of analytic conductor $C(\chi).$ We develop a regularized relative trace formula to prove a refined hybrid subconvex bound for $L(1/2,\pi\times\chi).$ In particular, we obtain the Burgess subconvex bound \begin{align*} L(1/2,\pi\times\chi)\ll_{\pi,F,\varepsilon}C(\chi)^{\frac{1}{2}-\frac{1}{8}+\varepsilon}, \end{align*} where the implied constant depends on $\pi,$ $F$ and $\varepsilon.$
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Squarefree numbers in short intervals: explicit and formalized
For intervals of length H = X^{1/5 - 2/90935 + ε}, the number of squarefree integers differs from (6/π²)H by at most an explicit constant times H X^{-ε/10^{25}}.
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