scholarly journals Another Proof of the Functional Equation for the Riemann Zeta Function

1994 ◽  
Vol 185 (1) ◽  
pp. 223-228 ◽  
Author(s):  
P.G. Rooney
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals Generalised Dirichlet series and Hecke's functional equation

1967 ◽  
Vol 15 (4) ◽  
pp. 309-313 ◽  
Author(s):  
Bruce C. Berndt
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
General Class ◽  
The Riemann Zeta Function ◽  
Entire Complex ◽  
Riemann Zeta ◽  
Image Position

The generalised zeta-function ζ(s, α) is defined bywhere α>0 and Res>l. Clearly, ζ(s, 1)=, where ζ(s) denotes the Riemann zeta-function. In this paper we consider a general class of Dirichlet series satisfying a functional equation similar to that of ζ(s). If ø(s) is such a series, we analogously define ø(s, α). We shall derive a representation for ø(s, α) which will be valid in the entire complex s-plane. From this representation we determine some simple properties of ø(s, α).

scholarly journals Value-Distribution of the Riemann Zeta-Function Along Its Julia Lines

2020 ◽  
Vol 20 (3-4) ◽  
pp. 389-401
Author(s):  
Jörn Steuding ◽  
Ade Irma Suriajaya
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Complex Number ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Distribution Of Values ◽  
Arbitrary Complex ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Average Behaviour

AbstractFor an arbitrary complex number $$a\ne 0$$ a ≠ 0 we consider the distribution of values of the Riemann zeta-function $$\zeta $$ ζ at the a-points of the function $$\Delta $$ Δ which appears in the functional equation $$\zeta (s)=\Delta (s)\zeta (1-s)$$ ζ ( s ) = Δ ( s ) ζ ( 1 - s ) . These a-points $$\delta _a$$ δ a are clustered around the critical line $$1/2+i\mathbb {R}$$ 1 / 2 + i R which happens to be a Julia line for the essential singularity of $$\zeta $$ ζ at infinity. We observe a remarkable average behaviour for the sequence of values $$\zeta (\delta _a)$$ ζ ( δ a ) .

A RELATION BETWEEN THE ZEROS OF TWO DIFFERENT L-FUNCTIONS WHICH HAVE AN EULER PRODUCT AND FUNCTIONAL EQUATION

2005 ◽  
Vol 01 (03) ◽  
pp. 401-429
Author(s):  
MASATOSHI SUZUKI
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Modular Forms ◽  
Functional Equations ◽  
Euler Product ◽  
The Riemann Zeta Function ◽  
Euler Products ◽  
Riemann Zeta ◽  
Special Case

As automorphic L-functions or Artin L-functions, several classes of L-functions have Euler products and functional equations. In this paper we study the zeros of L-functions which have Euler products and functional equations. We show that there exists a relation between the zeros of the Riemann zeta-function and the zeros of such L-functions. As a special case of our results, we find relations between the zeros of the Riemann zeta-function and the zeros of automorphic L-functions attached to elliptic modular forms or the zeros of Rankin–Selberg L-functions attached to two elliptic modular forms.

scholarly journals Some Identities and inequalities related to the Riemann zeta function

2019 ◽  
Vol 5 (2) ◽  
pp. 179-185
Author(s):  
Gregory Abe-I-Kpeng ◽  
Mohammed Muniru Iddrisu ◽  
Kwara Nantomah
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
New Inequalities ◽  
Trigonometric Identities

AbstractA new proof of Euler’s formular for calculating ζ(2k) is given. Some new inequalities and identities for ζ(2k + 1) have also been given. The Riemann’s functional equation together with trigonometric identities were used to establish the results.

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