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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A summation formula involving $ \sigma (N)$


Author: C. Nasim
Journal: Trans. Amer. Math. Soc. 192 (1974), 307-317
MSC: Primary 10A20
DOI: https://doi.org/10.1090/S0002-9947-1974-0337738-X
MathSciNet review: 0337738
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Abstract: The $ {L^2}$ theories are known of the summation formula involving $ {\sigma _k}(n)$, the sum of the kth power of divisors of n, as coefficients, for all k except $ k = 1$. In this paper, techniques are used to overcome the extra convergence difficulty of the case $ k = 1$, to establish a symmetric formula connecting the sums of the form $ \sum {\sigma _1}(n){n^{ - 1/2}}f(n)$ and $ \sum {{\sigma _1}} (n){n^{ - 1/2}}g(n)$, where $ f(x)$ and $ g(x)$ are Hankel transforms of each other.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0337738-X
Keywords: Fourier transform, Mellin transform, $ {L^2}$ class, Parseval theorem, convergence in mean square, Riesz summability
Article copyright: © Copyright 1974 American Mathematical Society