Harmonic numbers and harmonic sums

Harmonic, hyperharmonic and multiple harmonic sums, binomial-harmonic identities, non-integrality, harmonic congruences, Euler–Mascheroni bounds.

• 2-adic valuations of harmonic number differences
• Binomial identities for harmonic numbers
• Generalized harmonic numbers and the Gamma function
• Harmonic numbers and the Euler–Mascheroni constant
• Hyperharmonic Euler sums
• Hyperharmonic numbers and sums over primes
• Multiple harmonic sums
• Odd harmonic numbers
• Ramanujan's binomial–harmonic identity
• Ramanujan's harmonic sum identities

DefinitionHigher-order harmonic numbers#✓fully proved

For natural numbers \(n\) and \(r\), the harmonic number of order \(r\) is the rational number \(H_n^{(r)} = \sum_{k=1}^{n} 1/k^r\).

def higherHarmonic (n r : Nat) : Rat :=
  ∑ k ∈ Finset.Icc 1 n, (1 : Rat) / ((k : Rat) ^ r)

Used by 2

Used byCongruences for \(\binom{tp}{p}\) modulo \(p^5\), \(p^6\) and \(p^7\) · Congruences modulo \(p^7\) for \(\binom{\alpha p - 1}{p - 1} \binom{(\alpha + 1)p - 1}{p - 1}\) and \(\binom{2p - 1}{p - 1}^{-1}\)

TheoremSums of harmonic numbers starting from the second are not integers#✓fully proved

For every natural number \(n \ge 2\), the sum \(\sum_{\nu=2}^{n} H_\nu\) is not an integer, where \(H_\nu = \sum_{j=1}^{\nu} 1/j\) is the \(\nu\)-th harmonic number, viewed as a rational number.

The source proves this via its identity \(\sum_{\nu=2}^n H_\nu = (n+1)(H_n - 1)\) (equation (1.2)), reducing to the non-integrality of \((n+1) H_n\), in the spirit of Theisinger's theorem that \(H_n\) itself is never an integer for \(n > 1\).

theorem harmonic_partial_sum_not_int
    (n : ℕ) (hn : 2 ≤ n) :
    ¬ ∃ k : ℤ, (k : ℚ) = ∑ ν ∈ Finset.Icc 2 n, harmonic ν

Sources

• Horst Alzer and Man Kam Kwong, "Ramanujan and a Combinatorial Identity Involving Harmonic Numbers," Journal of Integer Sequences 29 (2026), Article 26.4.8, Theorem, lines 460–462, https://cs.uwaterloo.ca/journals/JIS/VOL29/Alzer/alzer22.tex

2 fully proved

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