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 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: BERNOULLI AND EULER POLYNOMIALS, RIEMANN ZETA FUNCTION 807 23.2. Riemann Zeta Function and Other Sums of Reciprocal Powers OD 23.2.1 f(~)=z k-s Ws>l 23.2.2 =n (l-p-+)-l %>l k= 1 P (product over all primes p ) . 23.2.3 * 23.2.4 23.2.5 where 23.2.6 23.2.7 23.2.8 23.2.9 23.2.10 product over all zeros p of {(s) with Wp>O. The contour C in the fourth formula starts at infinity on the positive real axis, circles the origin once in the positive direction excluding t,he points f 2 n i ~ for n=1, 2, . . ., and returns to the starting point. Therefore { ( s ) is regular for all values of s except for a simple pole at s = l with residue 1. Special Values 23.2.11 { ( O ) = - + 23.2.12 *See page 11. 23.2.13 f'(0) = -3 In 2~ 23.2.14 {(-2n) =O n = 1 , 2 , . . . n=1,2,. . . 23.2.15 {(l-2n) =-- B2n 2n n=1,2 ,... (2?p 2(2n)! 23.2.16 ((2n) = - lBznl 23.2.17 n=1,2,. . . Sums of Reciprocal Powers The sums referred to are m 23.2.18 {(n)=Z k-" n=2,3,. . . k = l 23.2.19 dn>=C k = l (-1)"1k-"=(1-21-n){(n) m n=1,2,. . . 23.2.20 X(n)=x (2k+l)-"=(l-2-"){(n) n=2,3,. . . OD k=O 23.2.21 m a(n) =C (-1)k(2k+1)-" n=1,2,. . . These sums can be calculated from the Bernoulli and Euler polynomials by means of the last two formulas for special values of the zeta function (note that T(l>=ln 2), and k =O 23.2.23 p(2) is known as Catalan's constant. Some other special values are 1 1 P2 23.2.24 {(2)=1+-+7+ . . . =- 22 3 6 1 1 Ir4 23.2.25 ((4)=1+-+-;i+ . . . =- 24 3 90 The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States.