The first moment of cusp form $L$-functions in weight aspect on average
Tom 181 / 2017
Acta Arithmetica 181 (2017), 197-208 MSC: Primary 11F11. DOI: 10.4064/aa8569-8-2017 Opublikowany online: 20 November 2017
We prove an asymptotic formula for the twisted first moment of central $L$-values in weight aspect on average. Our estimate of the error term allows extending the logarithmic length of the mollifier $\Delta $ up to $2$. The best previously known result, due to Iwaniec and Sarnak, was $\Delta \lt 1$. The proof is based on a representation formula for the error in terms of Legendre polynomials.