APPROXIMATE FUNCTIONAL EQUATION AND UPPER BOUNDS FOR THE BARNES DOUBLE ZETA-FUNCTION

AbstractWe obtain the approximate functional equation for the Rankin–Selberg zeta function in the critical strip and, in particular, on the critical line $\operatorname {Re} s= \frac {1}{2}$.

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Mean square of the Hurwitz zeta-function and other remarks

10.46298/hrj.2004.149 ◽

2004 ◽

Vol Volume 27 ◽

Author(s):

R Balasubramanian ◽

K Ramachandra

Keyword(s):

Functional Equation ◽

Zeta Function ◽

Analytic Continuation ◽

Hurwitz Zeta Function ◽

Complex Numbers ◽

Approximate Functional Equation ◽

The Riemann Zeta Function ◽

International Audience ◽

Approximate Function ◽

Riemann Zeta

International audience The Hurwitz zeta-function associated with the parameter $a\,(0< a\leq1)$ is a generalisation of the Riemann zeta-function namely the case $a=1$. It is defined by $$\zeta(s,a)=\sum_{n=0}^{\infty}(n+a)^{-s},\,(s=\sigma+it,\,\sigma>1)$$ and its analytic continuation. %In fact $$\zeta(s,a)=\sum_{n=0}^{\infty}\left((n+a)^{-s}-\int_{n}^{n+1}\frac{du}{(u+a)^s} \right)+\frac{a^{1-s}}{s-1}$$ gives the analytic continuation to $(\sigma>0)$. A repetition of this several times shows that $$\zeta-\frac{a^{1-s}}{s-1}$$ can be continued as an entire function to the whole plane. In $Re(s)\geq-1,\,t\geq2,\,\zeta(s,a)-a^{-s}=O(t^3)$ and by the functional equation (see \S2) it is $$O\left(\left(\frac{\vert s\vert}{2\pi}\right)^{\frac{1}{2}-Re(s)}\right)$$ in $Re(s)\leq-1,\,t\geq2$. From these facts In this paper, we deduce an `Approximate function equation' (see \S3), which is a generalisation of the approximate functional equation for $\zeta(s)$. Combining this with an important theorem due to van-der-Corput, we prove $$T^{-\frac{1}{3}}\int_{T}^{T+T^{\frac{1}{3}}} \vert\zeta(\frac{1}{2}+it)-a^{-\frac{1}{2}-it}\vert^2 dt <\!\!\!< (\log T)^3$$ uniformly in $a(0< a\leq1)$. From this we deduce similar results for quasi $L$-functions and more general functions. %Let $a_1, a_2,\ldots$, be any periodic sequence of complex numbers for which the sum over a period is zero. Let $b_1, b_2,\ldots$ be any sequence of complex numbers for which $\sum_{j=2}^{n}\vert b_j-b_{j-1}\vert+\vert b_n\vert\leq n^{\varepsilon}$ for every $\varepsilon>0$ and every $n\geq n_0(\varepsilon)$. Then we prove $$T^{-\frac{1}{3}}\int_{T}^{T+T^{\frac{1}{3}}} \vert\sum_{n=1}^{\infty}\frac{a_nb_n}{(n+a)^{\frac{1}{2}+it}}\vert^2\,dt\leq T^{\varepsilon}$$ for every $\varepsilon>0$ and every $T\geq T_0(\varepsilon)$. Here, as usual, $0<a\leq1$ and $T_0(\varepsilon)$ is independent of $a$.

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On the Mean-Square for the Approximate Functional Equation of the Riemann Zeta-Function

Tokyo Journal of Mathematics ◽

10.3836/tjm/1270128494 ◽

1993 ◽

Vol 16 (2) ◽

pp. 399-409 ◽

Cited By ~ 1

Author(s):

Isao KIUCHI

Keyword(s):

Functional Equation ◽

Zeta Function ◽

Riemann Zeta Function ◽

Mean Square ◽

Approximate Functional Equation ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

The Mean

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

Canadian Journal of Mathematics ◽

10.4153/cjm-2018-004-9 ◽

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.

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