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The first moment of cusp form $L$-functions in weight aspect on average

Volume 181 / 2017

Olga Balkanova, Dmitry Frolenkov Acta Arithmetica 181 (2017), 197-208 MSC: Primary 11F11. DOI: 10.4064/aa8569-8-2017 Published 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.


  • Olga BalkanovaDepartment of Mathematics and Statistics
    University of Turku
    Turku, 20014, Finland
  • Dmitry FrolenkovNational Research University Higher School of Economics
    Moscow, Russia
    Steklov Mathematical Institute
    Russian Academy of Sciences
    8 Gubkina st.
    Moscow, 119991, Russia

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