Wigert’s approximate functional equation and the Riemann zeta-function

1949 ◽  
Vol 16 (4) ◽  
pp. 547-552 ◽  
Author(s):  
Richard Bellman
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Approximate Functional Equation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Titchmarsh’s Method for the Approximate Functional Equations for , , and

2019 ◽  
Vol 71 (6) ◽  
pp. 1465-1493
Author(s):  
Jun Furuya ◽  
T. Makoto Minamide ◽  
Yoshio Tanigawa
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Functional Equations ◽  
Error Term ◽  
Approximate Functional Equation ◽  
The Riemann Zeta Function ◽  
Error Terms ◽  
Riemann Zeta

AbstractLet $\unicode[STIX]{x1D701}(s)$ be the Riemann zeta function. In 1929, Hardy and Littlewood proved the approximate functional equation for $\unicode[STIX]{x1D701}^{2}(s)$ with error term $O(x^{1/2-\unicode[STIX]{x1D70E}}((x+y)/|t|)^{1/4}\log |t|)$, where $-1/2<\unicode[STIX]{x1D70E}<3/2,x,y\geqslant 1,xy=(|t|/2\unicode[STIX]{x1D70B})^{2}$. Later, in 1938, Titchmarsh improved the error term by removing the factor $((x+y)/|t|)^{1/4}$. In 1999, Hall showed the approximate functional equations for $\unicode[STIX]{x1D701}^{\prime }(s)^{2},\unicode[STIX]{x1D701}(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$, and $\unicode[STIX]{x1D701}^{\prime }(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ (in the range $0<\unicode[STIX]{x1D70E}<1$) whose error terms contain the factor $((x+y)/|t|)^{1/4}$. In this paper we remove this factor from these three error terms by using the method of Titchmarsh.

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).

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