The Shifted Convolution Sums Of Divisor Functions And The Fourier Coefficients Of Cusp Forms

Let f(z)be a holomorphic cusp form of weight k for the modular group SL2(Z),and let ?f(n)be its normalized Fourier coefficients.Then f(z)has the Fourier expansionIn this thesis,we firstly consider the following shifted convolution sums of divisor functions and the Fourier coefficients of cusp forms:Our main results is for any h<<x1/3-?,we have D±(x)<<x 2/3+?,where the implied constant depend on f(z)and?.Let us also state the analogous result for the smoothed sum.For a smooth function w:R ? R,which is compactly supported in[1/2,1],define Dw±(x):=(?)d(n)?f(n±h)w(n/x)(h?1).Then we have for h<<x1/3-?where the implied constants depend on w,f(z)and ?,and ?=7/64.