# CERTAIN INTEGRAL REPRESENTATIONS FOR THE RIEMANN ZETA FUNCTION ζ(s) AT POSITIVE INTEGER ARGUMENT

Choi, Junesang

• Accepted : 2013.08.22
• Published : 2013.12.25
• 26 3

#### Abstract

We aim at presenting certain integral representations for the Riemann Zeta function ${\zeta}(s)$ at positive integer arguments by using some known integral representations of log ${\Gamma}(1+z)$ and ${\psi}(1+z)$.

#### Keywords

Gamma function;Riemann Zeta function;Hurwitz (or generalized) Zeta function;Psi (or Digamma) function;Polygamma functions;Euler-Mascheroni constant;Fa$\grave{a}$ di Bruno formula

#### References

1. J. Choi and H. M. Srivastava, Integral representations for the Euler-Mascheroni constant ${\gamma}$, Integral Transforms Spec. Funct. 21 (2010), 675-690. https://doi.org/10.1080/10652461003593294
2. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Corrected and Enlarged edition prepared by A. Jerey), Academic Press, New York, 1980; Sixth edition, 2000.
3. W. J. Kaczor and M. T. Nowak, Problems in Mathematical Analysis II, Continuity and Differentiation, American Mathematical Society, 2001.
4. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London, and New York, 2012.