Fractional part sums and divisor functions

Elementary methods are used to study sums of the form $\text{Σ}{}_{\mathrm{d\le x}}\text{{}\text{x}\text{d}\text{}}{}^{\mathrm{t}}$ for integers p and t, t > 0, where {x} denotes the fractional part of x. These sums are then used to study sums of the form $\text{Σ}{}_{\mathrm{d\le x}}\text{d}{}^{\mathrm{p}}\text{P}{}_{\mathrm{t}}\text{(}\text{x}\text{d}\text{)}$ for integers p and t, t > 0, where P_t(x) = B_t({x}) and B_t(x) are Bernoulli polynomials. some general results on sums of error terms are used to study sums of the form Σ_n≤xn^tσ_a(n) and Σ_n≤xE_t(n) for integers t and a, a ≥ 0, where σ_a(n) is the sum of the ath powers of the divisors of n and E_t(x) is the error term in the sum Σ_n≤xn^tσ_a(n).