## Integrals over polytopes, multiple zeta values and polylogarithms, and Euler’s constant.(English. Russian original)Zbl 1172.11028

Math. Notes 84, No. 4, 568-583 (2008); translation from Mat. Zametki 84, No. 4, 609-626 (2008); erratum Math. Notes 84, No. 6, 887 (2008).
The integral $$I_n=\int_0^1\int_{1-y}^1\frac{(-\ln xy)^n}{xy}\,dx\,dy$$ is shown to be a sum of multiple zeta values, $$I_n=n!\sum_{k=0}^n\zeta(n-k+2,\{1\}_k)$$, and $$I_n=P(\zeta(2),\zeta(3),\dots,\zeta(n+2))$$ for some polynomial with rational coefficients, which is explicitly given in the restatement of Theorem 1 on page 573. Combining this, it allows an expression for $$\zeta(n,\{1\}_k)$$ as a polynomial in $$\zeta(2),\dots,\zeta(n+k)$$ with rational coefficients, and the interesting relation $$\zeta(k+2,\{1\}_l)=\zeta(l+2,\{1\}_k)$$. The asymptotic expansion $$\frac{I_n}{n!}\sim\sum_{k=1}^\infty\binom{2k}{k}\frac1{k^{n+2}}$$ is derived. A formula for $$I_{-1}$$ is derived, and a similar formula for Euler’s constant $$\gamma$$. The authors discuss a number of examples.

### MSC:

 11M32 Multiple Dirichlet series and zeta functions and multizeta values 33B30 Higher logarithm functions
Full Text:

### Online Encyclopedia of Integer Sequences:

Decimal expansion of Euler’s constant (or the Euler-Mascheroni constant), gamma.

### References:

 [1] P. Cartier, “Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents,” Astérisque 282, Exp. no. 885, 137-173 (2002). · Zbl 1085.11042 [2] Dunham, W., Euler: The Master of Us All (1999), Washington, DC · Zbl 0951.01012 [3] S. Zlobin, On a Certain Integral over a Triangle, arXiv: math. NT/0511239. · Zbl 1058.11057 [4] F. Beukers, “A note on the irrationality of ζ(2) and ζ(3),” Bull. London Math. Soc. 11(3), 268-272 (1979). · Zbl 0421.10023 · doi:10.1112/blms/11.3.268 [5] K. S. Kölbig, J. A. Mignaco, and E. Remiddi, “On Nielsen’s generalized polylogarithms and their numerical calculation,” Nordisk Tidskr. Informationsbehandling (BIT) 10(1), 38-73 (1970). · Zbl 0196.17302 [6] J. M. Borwein, D. M. Bradley, and D. J. Broadhurst, “Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k,” Research paper R5, Electron. J. Combin. 4(2) (1997). · Zbl 0884.40004 [7] Waldschmidt, M., Multiple polylogarithms: An introduction, Proc. Int. Conf. held at Panjab University, Panjab University, Chandigarh, October 2-6, 2000, Basel · Zbl 1035.11033 [8] J. Sondow, “Criteria for irrationality of Euler’s constant,” Proc. Amer. Math. Soc. 131(11), 3335-3344 (2003). · Zbl 1113.11040 · doi:10.1090/S0002-9939-03-07081-3 [9] J. Sondow, “Double integrals for Euler’s constant and ln 4/π and an analog of Hadjicostas’s formula,” Amer. Math. Monthly 112(1), 61-65 (2005). · Zbl 1138.11356 · doi:10.2307/30037385 [10] J. Ser, “Sur une expression de la fonction ζ(s) de Riemann,” C. R. Acad. Sci. Paris Sér. I. Math. 182, 1075-1077 (1926). [11] J. Sondow, An Infinite Product for eγvia Hypergeometric Formulas for Euler’s Constant, γ, arXiv: math. CA/0306008 (2003). [12] J. Sondow, “A faster product for π and a new integral for ln π/2,” Amer. Math. Monthly 112(8), 729-734 (2005). · Zbl 1159.11328 [13] B. C. Berndt, Ramanujan’s Notebooks (Springer-Verlag, New York, 1985), Pt. I. · Zbl 0555.10001 [14] N. G. de Bruijn, Asymptotic Methods in Analysis (Dover Publ., New York, 1981). · Zbl 0556.41021 [15] S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, San Diego, 2000). · Zbl 0981.65001 [16] L. Lewin, Polylogarithms and Associated Functions (North-Holland Publ., New York, 1981). · Zbl 0465.33001 [17] J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek, “Special values of multiple polylogarithms,” Trans. Amer.Math. Soc. 353(3), 907-941 (2001). · Zbl 1002.11093 · doi:10.1090/S0002-9947-00-02616-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.