Abstract
Let σ( n) denote the sum of the divisors of n. Ramanujan proved that Σ 1≤n≤x σ 2(n) = 5 6 ζ(3)x 3 + E(x) , where E( x) = O( x 2 log 2 x). Using a theorem of Walfisz based on Weyl's inequality for exponential sums, we prove that E(x) = O(x 2 log 5 3 x) and also that Σ 1≤n≤x[ σ(n) n ] 2 = 5 2 ζ(3)x − 1 4 log 2x + O( log 5 3 x) .
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