Certain relationships among polygamma functions, Riemann zeta function and generalized zeta function

摘要

Many useful and interesting properties, identities, and relations for the Riemann zeta function ζ ( s ) and the Hurwitz zeta function ζ ( s , a ) have been developed. Here, we aim at giving certain (presumably) new and (potentially) useful relationships among polygamma functions, Riemann zeta function, and generalized zeta function by modifying Chen’s method. We also present a double inequality approximating ζ ( 2 r + 1 ) by a more rapidly convergent series. MSC:11M06, 33B15, 40A05, 26D07.