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Convolution sums arising from divisor functions

김대열|Li Yan|김애란
대한수학회 (The Korean Mathematical Society)|2013.03|대한수학회지 (Journal of the KMS)|제50권 제2호|pp. 331-360

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참고문헌

키워드

convolution sumsWeierstrass (x) functions
주제분류수학

초록

Let σs(N) denote the sum of the sth powers of the positivedivisors of a positive integer N and let σ~s(N) =∑d/N(-1)d-1ds with d,N, and s positive integers. Hahn [12] proved that 16∑k<N σ~1(k)σ~3(N - k) = -σ~5(N) + 2(N - 1)σ~3(N) + σ~1(N). In this paper, we give a generalization of Hahn's result. Furthermore, wend the formula ∑N-1k=1 σ~1(2n-mk)σ~3(2nN - 2nk) for m (0 ≤ m ≤ n).

Abstract

Let σs(N) denote the sum of the sth powers of the positivedivisors of a positive integer N and let σ~s(N) =∑d/N(-1)d-1ds with d,N, and s positive integers. Hahn [12] proved that 16∑k<N σ~1(k)σ~3(N - k) = -σ~5(N) + 2(N - 1)σ~3(N) + σ~1(N). In this paper, we give a generalization of Hahn's result. Furthermore, wend the formula ∑N-1k=1 σ~1(2n-mk)σ~3(2nN - 2nk) for m (0 ≤ m ≤ n).

참고문헌 (20)

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