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On the several identities of Riemann zeta-function. (English) Zbl 0755.11026

The author records various identities, as an example: If an integer n>4, then

24 a+b+c+d=n ζ(2a)ζ(2b)ζ(2c)ζ(2d)=(n+1)(2n+1)(2n+3)ζ(2n)-48nζ(2)ζ(2n-2)·(1)

Also, he points out a correction in the formula

96 a+b+c+d+e=n ζ(2a)ζ(2b)ζ(2c)ζ(2d)ζ(2e)==(n+1)(n+2)(2n+1)(2n+3)ζ(2n)-60n(n+1)ζ(2)ζ(2n-2)+144ζ 2 (2)ζ(2n-4)(2)

in which the constant 144 should be 216. The formula (2) was obtained by the reviewer in [Indian J. Pure Appl. Math. 18, 794-800 (1987; Zbl 0625.10031)]. This correction is trivial and it can be seen easily from the equation (3.1.6) of the above paper.

In the following paper [Indian J. Pure Appl. Math. 18, 891-895 (1987; Zbl 0635.10036)] much more has been proved by the reviewer jointly with K. Ramachandra. The method of the paper under review is not very different from the above mentioned paper.

11M06ζ(s) and L(s,χ)