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Register a new userEliminating oscillation in partial sum approximation of periodic function
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If we cannot obtain all terms of a series, or if we cannot sum up a series, we have to turn to the partial sum approximation which approximate a function by the first several terms of the series. However, the partial sum approximation often does not work well for periodic functions. In the partial sum approximation of a periodic function, there exists an incorrect oscillation which cannot be eliminated by keeping more terms, especially at the domain endpoints. A famous example is the Gibbs phenomenon in the Fourier expansion. In the paper, we suggest an approach for eliminating such oscillations in the partial sum approximation of periodic functions.
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Let $ xgeq 1 $ be a large number, let $ [x]=x-{x} $ be the largest integer function, and let $ sigma(n)$ be the sum of divisors function. This note presents the first proof of the asymptotic formula for the average order $ sum_{pleq x}sigma([x/p])=c_0xlog log x+O(x) $ over the primes, where $c_0>0$ is a constant. More generally, $ sum_{pleq x}sigma([x/(p+a)])=c_0xlog log x+O(x) $ for any fixed integer $a$.
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