Selberg sieve weights and sign changes of Kloosterman sums. I. Uniform asymptotics for shifted weights

Source abstract

We prove an asymptotic formula for correlations of divisor sums with quadratic cutoff functions shifted independently by $s,u\in[0,1/2]$. The error is $O_g(Y/(\log Y)^2)$, uniformly for $Y^{1/5}\leq R\leq Y^{1/3}$, including coincident shifts. The main term is an explicit bilinear form in the first and second derivatives of the cutoff functions. We obtain it by taking the residue in the outer Mellin variable and evaluating the remaining double integral by Laplace inversion. The present paper establishes the divisor-sum estimate. Its application to the sign-change problem, including the choice of the prime cutoff and the remaining parameters, is treated in Part II.