Geodesic
Representations of $\zeta(2n+1)$ and related numbers in the form of definite integrals and rapidly convergent series
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 494 (2020), pp. 48-52 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\zeta(s)$ and $\beta(s)$ be the Riemann zeta function and the Dirichlet beta function. The formulas for calculating the values of $\zeta(2m)$ and $\beta(2m-1)$ ($m=1,2,\dots$) are classical and well known. Our aim is to represent $\zeta(2m+1)$, $\beta(2m)$, and related numbers in the form of definite integrals of elementary functions and rapidly converging numerical series containing $\zeta(2m)$. By applying the method of this work, on the one hand, both classical formulas and ones relatively recently obtained by others researchers are proved in a uniform manner, and on the other hand, numerous new results are derived.
Keywords: integral representation of series sums, values of the Riemann zeta function at odd points, values of the Dirichlet beta function at even points
Mots-clés : Catalan's and Apéry's constants. 
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     title = {Representations of $\zeta(2n+1)$ and related numbers in the form of definite integrals and rapidly convergent series},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
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K. M. Mirzoev; T. A. Safonova. Representations of $\zeta(2n+1)$ and related numbers in the form of definite integrals and rapidly convergent series. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 494 (2020), pp. 48-52. http://geodesic.mathdoc.fr/item/DANMA_2020_494_a10/

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