An approximate functional equation for the zeta-function of Hecke in an imaginary quadratic field

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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THE $ \mathfrak{p}$-ADIC ZETA-FUNCTION OF AN IMAGINARY QUADRATIC FIELD AND THE LEOPOLDT REGULATOR

Mathematics of the USSR-Sbornik ◽

10.1070/sm1977v031n02abeh002295 ◽

1977 ◽

Vol 31 (2) ◽

pp. 151-158

Author(s):

M M Višik

Keyword(s):

Zeta Function ◽

Quadratic Field ◽

Imaginary Quadratic Field

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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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