Upper bounds for sums of powers of divisor functions

Abstract

Let ζ be the Riemann zeta-function and write ζ( s) 2 = Σ n >- 1 d z ( n) n − s for real s > 1, z > 1, so that d z ( n) is a generalized divisor function. We obtain good upper bounds for D z ( x, t) = Σ n ≤ x ( d z ( n)) t which are uniform in the real variables x, z, t when x ≥ 1, z > 1, and t > 0. We also derive sharp new estimates for the maximal order of d z ( n) which are uniform in both z and n. The proofs depend on precise uniform estimates for sums of p − σ ( p prime, σ > 0). The upper bounds for D z ( x, t) will be applied in a later paper to establish new results on the distribution of values of d z ( n).