Abstract
Abstract In this paper, we investigate the convolution sums ∑ ( a + b + c ) x = n a , ∑ a x + b y = n a b , ∑ a x + b y + c z = n a b c , ∑ a x + b y + c z + d u = n a b c d , where a , b , c , d , x , y , z , u , n ∈ N . Many new equalities and inequalities involving convolution sums, Bernoulli numbers and divisor functions have also been given. MSC:11A05, 33E99.
Highlights
Throughout this paper, N, Z, and C will denote the sets of positive integers, rational integers, and complex numbers, respectively
It is well known that Bk = Bk( ) are rational numbers
This completes the proof of this theorem
Summary
Throughout this paper, N, Z, and C will denote the sets of positive integers, rational integers, and complex numbers, respectively. Let n be an odd positive integer, and let f : Z → C be a complex-valued function. ≥ R–r– r+ – B (q + ) – B with M = q + This completes the proof of this theorem. Let M be an odd positive integer. = r – r – r – r + – r + – σ (m ), where we use the fact that r > r and a( n) = for n ∈ N
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