In this paper, we present a different proof of the well known recurrence formula for the Riemann zeta function at positive even integers, the integral representations of the Riemann zeta function at positive integers and at fractional points by means of a probabilistic approach.
The well known Riemann zeta function is defined by
which can be continued meromorphically to the whole complex -plane, except for a simple pole at , see [1,2,3] for details. Finding recurrence formulas and integral representations of the zeta function zeta function has become an important issue in complex analysis and number theory. One of the famous formulas is the following recursion formula for positive even integers
where and is the th Bernoulli number. Here is the set of positive integers. Several new proofs to (1) can be found in [4,5,6,7]. A new parameterized series representation of zeta function is derived in . However, no similar closed-form representation of at odd integers or fractional points can be found in literature. The Riemann zeta function for positive odd integer arguments can be expressed by series and integrals. One possible integral expression is established by  as follows
where are Bernoulli polynomials defined by the generating function 
The Bernoulli numbers are well-tabulated (see, for example, ):
More lists of Bernoulli numbers and their estimation can be found in the recent work by Qi .
The zeta function has many integral representations, one of which is the following  (P.172) (note that there is an extra 2 in (51) of  (P.172):
The aim of this note is to present a new proof of (1) for and deduce the integral representations for and . The proofs are based on the characteristic function and the moment generating function of logistic, half-logistic and elliptical symmetric logistic distributions in probability theory and mathematical statistics.
2. The Main Results and Their Proofs
In this section we present a new proof to the following results by using a probabilistic method. To the best of our knowledge, the result (5) is new.
For Riemann’s zeta function, we have
whereis theth Bernoulli number.
To prove the proposition, we need the following three lemmas.
We assume that random variablehas the standard logistic distribution with the probability density function (pdf)
Then the moment generating function (mgf) ofis given by
where is defined by (7). Taking th and th derivatives of the two functions with respect to , we get
Note that for any real , and thus for any real and any positive integers . In particular, . However, , and from we have
which concludes the proof of (5).
Finally, we prove (6). Using (12) one has
Taking the th derivative of both sides of (14) with respect to and then setting yields the desired result. ☐
These authors contributed equally to this work.
The research was supported by the National Natural Science Foundation of China (No. 11571198)
The authors wish to acknowledge the comments and suggestions made by the anonymous referees that helped in improving this version of the paper. The research was supported by the National Natural Science Foundation of China (No. 11571198)
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Choi, J.; Cho, Y.J.; Srivastava, H.M. Series involving the zeta function and multiple Gamma functions. Appl. Math. Comput.2004, 159, 509–537. [Google Scholar] [CrossRef]
Gradshteyn, I.S.; Ryzhik, I.M. Tables, Integrals, Series, and Products; Academic: New York, NY, USA, 1980. [Google Scholar]
Srivastava, H.M. Certain classes of series associated with the zeta and related functions. Appl. Math. Comput.2003, 141, 13–49. [Google Scholar] [CrossRef]
Titchmarsh, E.C.; Heath-Brown, D.R. The Theory of the Riemann Zeta-function, 2nd ed.; Oxford University Press: Oxford, UK, 1986. [Google Scholar]
De Amo, E.; Díaz Carrillo, M.; Fernández-Sánchez, J. Another proof of Euler’s formula for ζ(2k). Proc. Am. Math. Soc.2011, 139, 1441–1444. [Google Scholar] [CrossRef]
Arakawa, T.; Ibukiyama, T.; Kaneko, M. Bernoulli Numbers and Zeta Functions; Springer Monographs in Mathematics; Springer: Tokyo, Japan, 2014. [Google Scholar]
Ribeiro, P. Another proof of the famous formula for the zeta function at positive even integers. Am. Math. Mon.2018, 125, 839–841. [Google Scholar] [CrossRef]
Qi, F. A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers. J. Comput. Appl. Math.2019, 351, 1–5. [Google Scholar] [CrossRef]
Srivastava, H.M.; Choi, J. Zeta and Q-Zeta Functions and Associated Series and Integrals; Elsevier Inc.: New York, NY, USA, 2012. [Google Scholar]
Johnson, N.L.; Kotz, S.; Balakrishnan, N. Continuous Univariate Distributions, 2nd ed.; Wiley: New York, NY, USA, 1995; Volume 2. [Google Scholar]
Ghosh, M.; Choi, K.P.; Li, J. A commentary on the logistic distribution. In The Legacy of Alladi Ramakrishnan in the Mathematical Sciences; Alladi, K., Klauder, R., Rao, R., Eds.; Springer Science+Business Media: Berlin, Germany, 2010. [Google Scholar]
Widder, D.V. The Laplace Transform, 8th ed.; Princeton University Press: Princeton, NJ, USA, 1972. [Google Scholar]