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We establish a relative trace formula on $\mathrm{GL}(n+1)$ weighted by cuspforms on $\mathrm{GL}(n)$ over number fields. The spectral side is a weightedaverage of Rankin-Selberg $L$-functions for$\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over the \textit{full spectrum}, and thegeometric side consists of Rankin-Selberg $L$-functions for$\mathrm{GL}(n)\times\mathrm{GL}(n),$ and certain explicit holomorphicfunctions. The formula yields new results towards central $L$-values for$\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ (over number fields): the second momentevaluation, and simultaneous nonvnaishing in the level aspect.

Relative Trace Formula and $L$-functions for $\mathrm{GL}(n+1)\times \mathrm{GL}(n)$