Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(cm-2,…, c1, n): Let λ(n) be either the von Mangoldt function Λ(n) or the k-th divisor function τk(n): We consider averages of shifted convolu...Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(cm-2,…, c1, n): Let λ(n) be either the von Mangoldt function Λ(n) or the k-th divisor function τk(n): We consider averages of shifted convolution sums of the type Σ|h|≤H |ΣX<n≤2XAf (1,…, 1, n+h)λ(n)|^2. We succeed in obtaining a saving of an arbitrary power of the logarithm, provided that X^8/33+ε≤H≤X^1-ε.展开更多
文摘Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(cm-2,…, c1, n): Let λ(n) be either the von Mangoldt function Λ(n) or the k-th divisor function τk(n): We consider averages of shifted convolution sums of the type Σ|h|≤H |ΣX<n≤2XAf (1,…, 1, n+h)λ(n)|^2. We succeed in obtaining a saving of an arbitrary power of the logarithm, provided that X^8/33+ε≤H≤X^1-ε.
