Abstract
We use everywhere-convergent series for the height 1 multiple zeta functions \(\zeta (s,1,\ldots ,1)\) to determine the singular parts of their Laurent series at each of their poles, and give an expression for each first “Stieltjes constant” (i.e., the linear Laurent coefficient) as series involving the Bernoulli numbers of the second kind, generalizing the classical Mascheroni series for Euler’s constant \(\gamma \). The first Stieltjes constants at \(s=1\) and at \(s=0\) are then interpreted in terms of the Ramanujan summation of multiple harmonic star sums \(\zeta ^\star (1,\ldots ,1)\).
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Acknowledgements
The author thanks the reviewers of this manuscript for their helpful comments, and Marc-Antoine Coppo for helpful conversations. All numerical computation was done using the PARI-GP calculator created by C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier.
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Young, P.T. Global series for height 1 multiple zeta functions. European Journal of Mathematics 9, 99 (2023). https://doi.org/10.1007/s40879-023-00695-0
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DOI: https://doi.org/10.1007/s40879-023-00695-0