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arXiv · 2107.01030

Sum of Divisors Function And The Largest Integer Function Over The Shifted Primes

N. A. Carella
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Abstract

Let $ x\geq 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_{p\leq x}\sigma([x/p])=c_0x\log \log x+O(x) $ over the primes, where $c_0>0$ is a constant. More generally, $ \sum_{p\leq x}\sigma([x/(p+a)])=c_0x\log \log x+O(x) $ for any fixed integer $a$.

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N. A. Carella. 2021-06-30. Sum of Divisors Function And The Largest Integer Function Over The Shifted Primes. https://arxiv.org/abs/2107.01030

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