Number Theory

    

On the Harmonic Distribution of Addition Chains

Authors: Theophilus Agama

In this note, we study the harmonic distribution of addition chains of a given length. If $1,2,ldots,s_{delta(n)-1},s_{delta(n)}=n$ is an addition chain producing $n$ and of length $delta(n)$, with associated sequence of generators begin{align}1+1,s_2=a_2+r_2,ldots,s_{delta(n)-1}=a_{delta(n)-1}+r_{delta(n)-1},s_{delta(n)}=a_{delta(n)}+r_{delta(n)}=nonumberend{align}then $$sum limits_{l=1}^{delta(n)}frac{1}{s_l}=frac{3}{2}+frac{delta(n)}{n+1}+sum limits_{l=3}^{delta(n)}sum limits_{v=1}^{infty}frac{1}{(n+1)^{v+1}}bigg(sum limits_{j=l}^{delta(n)}r_jbigg)^{v}+O(frac{1}{n})$$ where $sum limits_{j=l}^{delta(n)}r_j<n-1$ is the sum of the regulators in the generator of the chain for each $3leq lleq delta(n)$.

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[v1] 2025-01-07 21:58:31

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