2Values of the Riemann zeta function at integers. The Zeta function is a very important function in mathematics. The Prime Number Theorem, Hardy’s theorem on the Zeros of ζ(s), and Hamburger’s theorem are the princi-pal results proved here. Speciﬁ-cally, the expansion given is … This allows an asymptotic estimation of a quantity which came up in the theory of the Riemann zeta-function. Download file PDF Read file. While it was not created by Riemann, it is named after him because he was able to prove an important relationship between its zeros and the distribution of the prime numbers. • The poles of Γ/2+1 are cancelled by the trivial zeros of the zeta function, at s = -2, -4, -6, etc. The function was rst studied by Leonhard Euler as a function of real variable and then extended by Bernhard Riemann to the entire complex plane. Thus is an entire function with zeros precisely at the nontrivial zeros of The Riemann zeta function ~(s) is defined by the series En>l n-S for any complex number with Re (s) > 1. The exposition is self-contained, a function of a complex variable s= x+ iyrather than a real variable x. The Riemann Zeta Function is a function of complex variable which plays an important role in analytic number theory and prime number theorem. tion to the theory of the Riemann Zeta-function for stu-dents who might later want to do research on the subject. More precisely, they rely on three key ingredients: a numerical veri cation of the Riemann Hypothesis (RH), an explicit zero-free region, and explicit bounds for the number of zeros in the critical strip up to a xed height T. In 1986, van de Lune et al. [34] established that RH had been veri ed for all zeros %verifying jIm%j H 2 De nition of zeta function and Functional Equa-tion His result is critical to the proof of the prime number theorem. mann zeta function. However, the formula (2) cannot be applied anymore if the real part • The pole of zeta at s = 1 is cancelled by (−1). We will use Riemann’s definition of (), which has the advantage that it has no poles. Some new properties of … Moreover, in 1859 Riemann gave a formula for a unique (the so-called holo-morphic) extension of the function onto the entire complex plane C except s= 1. In this paper, a relationship to the Riemann zeta function [Ed74] is noted, allowing the easy derivation of a series expansion of the zeta function in terms of the Pochham-mer symbols (the falling factorials), or equivalently the binomial coefﬁcients. The theory of the zeta function implies that its definition can be extended (not by the same series, of course) to all values of s other than s = 1.