5

Let $H_n^{(p)} = 1^{-p} + 2^{-p} + \dots + n^{-p}$ be the generalized harmonic number.

Consider the alternating series:

$$ T = \sum_{n=1}^\infty \frac{(H_n^{(2)})^2}{n^3} (-1)^{n-1}. $$

The non-alternating version is known to be a rational combination of $\zeta(7)$ and $\zeta(2)\zeta(5)$.

Question: Is there a known closed form for $T$ in terms of classical constants (zeta values, polylogarithms at 1/2, etc.)?

Numerical value (2000 terms, Mathematica):

0.395211126203650535525054...

A PSLQ search against $\zeta(7), \zeta(2)\zeta(5), \zeta(3)\zeta(4), \mathrm{Li}_7(1/2)$ found no small relation. I couldn't find this form in existing literature on Euler sums.

RobPratt
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NotEinstein
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1 Answers1

1

Not an answer, just addressing the value of $T$...

(* Mathematica 14.2.1.0 ... *)
Table[N[Sum[
      (-1)^(n - 1) HarmonicNumber[n, 2]^2/n^3 , 
      {n, 1, 2 10^k}], 
    30], 
  {k, 0, 4}]


(* Output: *)
{0.804687500000000000000000000000,
0.852316238062022360526613185413,
0.852463645913350236905169665307,
0.852463812575442867194084443613,
0.852463812744157283410448740631}

The Question's value, 0.3952111262... is implausible. (These are all even-length partial sums, so are all lower bounds. The one-term-more upper bounds are

{0.873303183584819387288523090992, 
 0.852592124151935190103137344785, 
 0.852463977108770115588729168298, 
 0.852463812912956922014943900234, 
 0.852463812744495438135417155194}

)

Eric Towers
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