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Special value formula for the twisted triple product $L$-function and an application to the restricted $L^2$-norm problem
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abstract
We establish explicit Ichino's formulae for the central values of the triple product $L$-functions with emphasis on the calculations for the real place. The key ingredient for our computations is Proposition 6.8 which generalizes a result of Michel-Venkatesh. As an application we prove the optimal upper bound of a sum of restricted $L^2$-norms of the $L^2$-normalized newforms on certain quadratic extensions with prime level and bounded spectral parameter following the methods of Blomer.
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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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