Zeta and Egyptian Fractions

What values of ξ(−n) \xi (-n)\quad (where n = odd number i.e, 1, 3, 5, 7) ..... can not be represented by Egyptian fractions or Egyptian fractions - x (where x is any integer) form. I am only considering Egyptian fraction with length greater than 1.

  • For instance ξ(−1) \xi (-1)\quad = -1/12 = [(1/2)+(1/3)+(1/12)] - 1
  • ξ(−17) \xi (-17)\quad = 1/2+1/3+1/9+1/625+1/816137+1/1046695702500 - 4
  • ξ(−11) \xi (-11)\quad = 691/32760 = (1/48)+(1/3855)+(1/16838640)

However if you see ξ(−3) \xi (-3)\quad = 1/120 or ξ(−7) \xi (-7)\quad = 1/240 they can not be represented as Egyptian faction of length > 1 Also note that since we are only considering Egyptian fraction with length greater than 1 ξ(−33) \xi (-33)\quad which equals (1/12) -12635724796 also becomes part of this sequence

So If we start creating a sequence S(n) for which this is not possible then 3, 7 and 33 are first few candidates. I would like to learn more about this sequence. Also is there any way to know, for what values of n, ξ(−n) \xi (-n)\quad will be positive?

#NumberTheory #Zeta

Note by Vikram Pandya
5 years, 9 months ago

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