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Find the limsup!

Source: CIIM 2023 - Problem 4

September 19, 2023

limit superiorlimitsum of divisorsreal analysis

Problem Statement

For a positive integer

n

,

\sigma(n)

denotes the sum of the positive divisors of

n

. Determine

\limsup\limits_{n\rightarrow \infty} \frac{\sigma(n^{2023})}{(\sigma(n))^{2023}}

Note: Given a sequence (

a_n

) of real numbers, we say that

\limsup\limits_{n\rightarrow \infty} a_n = +\infty

if (

K > C

) is not upper bounded, and, otherwise,

\limsup\limits_{n\rightarrow \infty} a_n

is the smallest constant

C

such that, for every real

K > C

, there is a positive integer

N

with

a_n < K

for every

n > N

.

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