Geodesic
The first moment of cusp form $L$-functions in weight aspect on average
Olga Balkanova 1   ; Dmitry Frolenkov 2
1 Department of Mathematics and Statistics University of Turku Turku, 20014, Finland
2 National Research University Higher School of Economics Moscow, Russia and Steklov Mathematical Institute Russian Academy of Sciences 8 Gubkina st. Moscow, 119991, Russia
Acta Arithmetica, Tome 181 (2017) no. 3, pp. 197-208 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/aa8569-8-2017
Keywords: prove asymptotic formula twisted first moment central l values weight aspect average estimate error term allows extending logarithmic length mollifier delta best previously known result due iwaniec sarnak delta proof based representation formula error terms legendre polynomials

Olga Balkanova  1   ; Dmitry Frolenkov  2

1 Department of Mathematics and Statistics University of Turku Turku, 20014, Finland
2 National Research University Higher School of Economics Moscow, Russia and Steklov Mathematical Institute Russian Academy of Sciences 8 Gubkina st. Moscow, 119991, Russia 
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     title = {The first moment of cusp form $L$-functions in weight aspect on average},
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Olga Balkanova; Dmitry Frolenkov. The first moment of cusp form $L$-functions in weight aspect on average. Acta Arithmetica, Tome 181 (2017) no. 3, pp. 197-208. doi: 10.4064/aa8569-8-2017

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