Abstract. The famous multiplicitous sum of reciprocal powers Σ k Σs k−s = 1 = Σ V Σs (τ(s)-1)/vs where V is the set of powers, v its elements and τ(s) is the number of divisors function over all positive integer exponents s > 1.
Finding an authoritative historical account of the double-sum equality ΣkΣs k−s = 1 where k,s=2→∞, is difficult, in part because this double sum may be expressed in many other ways, a property stemming from its equivalence to unity. Sometimes attributed to Leibniz and/or Huygens, or even Goldbach via Euler, but the earliest variant I could find was Mengoli's sum of the inverses of the triangular numbers, Σ(2Tk)−1=1 where k=1→∞ and Tk=k(k+1)/2.
Although obvious from the notation, the 'reciprocals of the powers' sum is typically noted as "including duplicates" or "with multiplicity", primarily to avoid confusion with the variant Goldbach did prove, Σk,s >1 (ks-1)−1 = 1, where each term corresponds to a unique perfect power. In the paper, I have solved a number theoretic relation that codifies the multiplicity of the first double sum and demonstrates how all of these variants can be transformed into each other (as each is equivalent to unity) using the geometric series identity and some splitting and recombining of summation terms.
I promulgate a notation system over the sets V with elements v, and W with elements w, to represent the non-power integers and the perfect powers, respectively. The notation not only obviates the need to attach notes to formulae in order to clarify the state of their multiplicity, it seamlessly allows expressions over V or W to be equated directly with each other and with sums indexed over the positive integers in the standard way.
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Euler, L., Various Observations About Infinite Series, Commentarii academiae scientarum Petropolitanae 9 (1737), 1744, pp. 160-188. Comment # 72 from Enestroemiani's Index. Translated from Latin by Peligrí Viader and Lluís Bibiloni and Pelegrí Viader Jr., via The Euler Project.
Sloane, N. J. A.,Perfect Powers, Sequence A001597, The On-Line Encyclopedia of Integer Sequences, A001597, 2006 http://www.research.att.com/~njas/sequences/A001597
Viader, P., Bibiloni, L., and Paradís, J., On a Series of Goldbach and Euler, American Mathematical Monthly, March, 2006
Boland, D., The Zeta Function by the Non-Powers Sketches the Riemann Hypothesis, Apartment of Mathematics, Imathination.org, 2006, http://www.imathination.org/docs/zeta_by_v.pdf
Sloane, N. J. A., Decimal Expansion of Sum of Reciprocal Perfect Powers Without Duplication, Sequence A072102, The On-Line Encyclopedia of Integer Sequences, A072102, 2006, http://www.research.att.com/~njas/sequences/A072102
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