Let f be a holomorphic newform of prime level p and trivial nebentypus. For $p^{1+\epsilon }\ll M\ll p^{3-\epsilon }$, and $0<|u|\ll M/p$ we prove that$\sum _{m=1}^{\infty }{\lambda }_f(m){\lambda }_f(m+pu)F\left(\frac{m}{M}\right)\ll p^{1/4+\epsilon }{M}^{3/4}$ where F is a compactly supported smooth bump function.

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关于平移卷积求和问题

令f是素数p和平凡 nebentypus的全纯新形式。为了${磷}^{1+\epsilon }\ll 米\ll {磷}^{3-\epsilon }$， 和 $0<|你|\ll 米/磷$ 我们证明$\sum _{米=1}^{\infty }{\lambda }_F(米){\lambda }_F(米+磷你)F\left(\frac{米}{米}\right)\ll {磷}^{1/4+\epsilon }{米}^{3/4}$其中F是紧凑支持的平滑凹凸函数。