## Abstract

We determine the values of several infinite series involving the Riemann zeta function. In particular, degree one rationally weighted summands involving the Riemann function evaluated at even numbers give a finite sum involving only the Riemann function evaluated at odd numbers.

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## Notes

Actually \(\lim _{n\rightarrow \infty }(\zeta (2n)-1) 4^{n}=1\). In fact,

$$\begin{aligned} 4^n(\zeta (2n)-1) =\sum _{k=2}^\infty \left( \frac{2}{k}\right) ^{2n}=1 + \sum _{k=3}^\infty \left( \frac{2}{k}\right) ^{2n}. \end{aligned}$$Now the convergence of \(\sum _{k=3}^\infty \left( \frac{2}{k}\right) ^{2x}\) is uniform in \(x\ge 1\) (due to the converging majorant \(\sum _{k=3}^\infty \left( \frac{2}{k}\right) ^{2}\)). Hence \(\lim _{x\rightarrow \infty }\sum _k=\sum _k\lim _{x\rightarrow \infty }\). The assertion follows, as each summand goes to 0 as \(n\rightarrow \infty \).

That formula holds of course, too, if we replace 2

*m*by*m*; a fact we need in the proof of Theorem 3.2.As usual, the empty sum is defined to be the number 0.

## References

Choi, J., Srivastava, H.M.: Sums associated with the zeta function. J. Math. Anal. Appl.

**206**, 103–120 (1997)Fischer, W., Lieb, I.: Funktionentheorie. Vieweg & Sohn, Braunschweig (1988)

Johnson, W.W.: Note on the numerical transcendents \(S_n\) and \(s_n=S_n-1\). Bull. Amer. Math. Soc.

**2**(12), 477–482 (1906)Mortini, R., Rupp, R.: Extension Problems and Stable Ranks-A Space Odyssey. Birkhäuser/Springer, Cham (2021)

Omarjee, M.: Problem 2167. Math. Mag.

**96**, 190 (2023)Srivastava, H.M.: Sums of certain series of the Riemann zeta function. J. Math. Anal. Appl.

**134**, 129–140 (1988)Srivastava, H.M.: A unified presentation of certain classes of series of the Riemann zeta function. Riv. Mat. Univ. Parma (4)

**14**, 1–23 (1988)Verma, D.P., Kaur, A.: Summation of some series involving Riemann zeta function. Indian J. Math.

**25**, 181–184 (1983)How to evaluate \(\sum _{n=1}^\infty \frac{\zeta (2n)-1}{n+1}\) directly. https://math.stackexchange.com/questions/3753490/how-to-evaluate-sum-n-1-infty-frac-zeta-2n-1n1-directly/3754173

## Acknowledgements

We thank Roberto Tauraso for providing us reference [6].

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Mortini, R., Rupp, R. Sums of infinite series involving the Riemann zeta function.
*Arch. Math.* **123**, 163–172 (2024). https://doi.org/10.1007/s00013-024-02008-7

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DOI: https://doi.org/10.1007/s00013-024-02008-7