## Pacific Journal of Mathematics

### Multiple harmonic series.

Michael E. Hoffman

#### Article information

Source
Pacific J. Math. Volume 152, Number 2 (1992), 275-290.

Dates
First available in Project Euclid: 8 December 2004

http://projecteuclid.org/euclid.pjm/1102636166

Mathematical Reviews number (MathSciNet)
MR1141796

Zentralblatt MATH identifier
0763.11037

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 40A05: Convergence and divergence of series and sequences

#### Citation

Hoffman, Michael E. Multiple harmonic series. Pacific J. Math. 152 (1992), no. 2, 275--290. http://projecteuclid.org/euclid.pjm/1102636166.

#### References

• [4] Partial proof of the duality conjecture. We shall prove the duality conjecture for sequences (i9 1, ... , 1) with i\ > 1. We use the following theorem of L. J. Mordell [6]. THEOREM 4.1 (Mordell). For positive integer k and any a> --k, y! =fc,f (-'(a-i\ >,i 2*(i + + * + )"^j /!(/+ 1)*+* \iJ' th ^ 1 From this we deduce the following result.
• [5] Evidence for the conjectures. Forthecomputations of thissec- tion, thefollowing result will be useful. THEOREM 5.1. Let i\, i2, ... , ik be any sequence of positive inte- gers with i\ > 1. Then k i-2 A(ii,.,i-i,i-J,J+l,i+i,..,ik) \<l<kj=O Proof.Bymultiplying series, we have (1)A(i + l, i2,...,ik) +A{iu1, 2, ... , ik) + A{i, i2+ 1 ,... , ik) + ''' +Ah , *2> . - > * * , 1) ^ 2- - - ^7
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