Monotonic Divergent Series of Number Theoretic Functions

SPECIES

Monotonic divergent series of numbers of divisors of n

$\displaystyle \sum_{n\geq1} \sigma_0(n) = 1+2+2+3+2+4+2+4+\cdots = \dfrac{1}{4}$ (unstable)

Monotonic divergent series of sums of divisors of n

$\displaystyle \sum_{n\geq1} \sigma_1(n) = 1+3+4+7+6+12+8+15+\cdots = \dfrac{1}{24}$ (unstable)

Monotonic divergent series of numbers of prime divisors

$\displaystyle \sum_{n\geq1} \Omega(n) = 0+1+1+2+1+2+1+3+2+\cdots = -\dfrac{1}{8}$ (unstable)

Monotonic divergent series of numbers of distinct prime divisors

$\displaystyle \sum_{n\geq1} \omega(n) = 0+1+1+1+1+2+1+1+1+\cdots = \dfrac{1}{4}$ (unstable)

Monotonic divergent series of Euler totient numbers

$\displaystyle \sum_{n\geq1} \varphi(n) = 1+1+2+2+4+2+6+4+\cdots = \dfrac{1}{6}$ (unstable)

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