DOIAbstract
AbstractIt is proved that (for every ε > 0) ∑n⩽T13 ∑n<Tn12 namb Bk({Tnm}) = O(T(a+b+1)3ϵ) (where {·} denotes the fractional part and Bk the Bernoulli polynomial of order k) under the suppositions that k ≥ 2 and 2a − 1 ≥ b ≥ 1. If (∗) were true for k = 1, a = b = 0, then Piltz' divisor problem (for n = 3) would be readily solved. This is an analog to a conjecture formulated by S. Chowla and H. Walum in 1963 and settled in the affirmative (under suitable suppositions) quite recently by S. Kanemitsu and R. Sita Rama Chandra Rao
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