Part 1 of 2

• Alder's totient function
• Average and divisibility of the divisor sum
• Dedekind psi function
• Divisor-power sums
• Divisors in short intervals
• Euler's function and divisor sums along linear recurrences
• Gaps between divisors
• Jordan's totient function
• Liouville's divisor-function identity
• Nicol numbers
• Parity of divisor counts
• Pillai's gcd-sum function
• Ramanujan sums
• Residues of rounded quotients n/k
• Totient of the sum of divisors
• Unitary divisors
• Biunitary divisors
• Regular integers modulo n
• Values of n + φ(n)

TheoremFord's theorem on totient multiplicities#●ready to prove

For every natural number \(k \ge 2\) there is a natural number \(m\) such that exactly \(k\) natural numbers \(x\) satisfy \(\varphi(x) = m\), where \(\varphi\) is Euler's totient function.

theorem_wanted ford_totient_multiplicity :
    ∀ k : ℕ, 1 < k → ∃ m : ℕ, Set.ncard {x : ℕ | Nat.totient x = m} = k

Sources

• Kevin Ford, The Number of Solutions of phi(x) = m, Annals of Mathematics 150 (1999), DOI 10.2307/121103. Also: https://en.wikipedia.org/wiki/Euler%27s_totient_function

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