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Functional equation for the zeta function: Difference between revisions

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== Introduction ==
The '''functional equation for Riemann zeta function''' is a result due to analytic continuation of [[Riemann zeta function]]:
The '''functional equation for Riemann zeta function''' is a result due to analytic continuation of [[Riemann zeta function]]:


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</cmath>
</cmath>


=== From <math>\sigma>1</math> to <math>\sigma>-1</math> ===
=== A formula for <math>\zeta(s)</math> in <math>-1<\sigma<0</math> ===


In this article, we will use the common convention that <math>s=\sigma+it</math> where <math>\sigma,t\in\mathbb R</math>. As a result, we say that the original [[Dirichlet series]] definition <math>\zeta(s)\triangleq\sum_{k=1}^\infty{1\over k^s}</math> converges only for <math>\sigma>1</math>. However, if we were to apply Euler-Maclaurin summation on this definition, we obtain
In this article, we will use the common convention that <math>s=\sigma+it</math> where <math>\sigma,t\in\mathbb R</math>. As a result, we say that the original [[Dirichlet series]] definition <math>\zeta(s)\triangleq\sum_{k=1}^\infty{1\over k^s}</math> converges only for <math>\sigma>1</math>. However, if we were to apply Euler-Maclaurin summation on this definition, we obtain
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<cmath>
<cmath>
\begin{aligned}
\begin{align*}
\int_1^\infty{B_1(x)\over x^{s+1}}\mathrm dx
\int_1^\infty{B_1(x)\over x^{s+1}}\mathrm dx
={B_2(x)\over2x^{s+1}}+{s+1\over2}\int_1^\infty{B_2(x)\over x^{s+2}}\mathrm dx
&=\left.{B_2(x)\over2x^{s+1}}\right|_1^\infty+{s+1\over2}\int_1^\infty{B_2(x)\over x^{s+2}}\mathrm dx \\
\end{aligned}
&={B_2\over2}+{s+1\over2}\int_1^\infty{B_2(x)\over x^{s+2}}\mathrm dx
\end{align*}
</cmath>
</cmath>

Revision as of 02:22, 13 January 2021

The functional equation for Riemann zeta function is a result due to analytic continuation of Riemann zeta function:

\[\zeta(s)=2^s\pi^{s-1}\sin\left(\pi s\over2\right)\Gamma(1-s)\zeta(1-s)\]

Proof

Two useful identities

There are multiple proofs for the functional equation for Riemann zeta function, and this page presents a light-weighted approach which merely relies on the Fourier series for the first periodic Bernoulli polynomial that

\[B_1(x)\triangleq\{x\}-\frac12=-\sum_{n=1}^\infty{\sin(2\pi nx)\over\pi n}\]

A formula for $\zeta(s)$ in $-1<\sigma<0$

In this article, we will use the common convention that $s=\sigma+it$ where $\sigma,t\in\mathbb R$. As a result, we say that the original Dirichlet series definition $\zeta(s)\triangleq\sum_{k=1}^\infty{1\over k^s}$ converges only for $\sigma>1$. However, if we were to apply Euler-Maclaurin summation on this definition, we obtain

\[\zeta(s)=\frac12+{s\over s-1}-s\int_1^\infty{B_1(x)\over x^{s+1}}\mathrm dx\]

in which we can extend the ROC of the latter integral to $\sigma>-1$ via integration by parts:

\begin{align*} \int_1^\infty{B_1(x)\over x^{s+1}}\mathrm dx &=\left.{B_2(x)\over2x^{s+1}}\right|_1^\infty+{s+1\over2}\int_1^\infty{B_2(x)\over x^{s+2}}\mathrm dx \\ &={B_2\over2}+{s+1\over2}\int_1^\infty{B_2(x)\over x^{s+2}}\mathrm dx \end{align*}

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