Difference between revisions of "Functional equation for the zeta function"
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| − | The '''functional equation for Riemann zeta function''' is a result due to analytic continuation of Riemann zeta function: | + | == Introduction == |
| + | |||
| + | The '''functional equation for Riemann zeta function''' is a result due to analytic continuation of [[Riemann zeta function]]: | ||
<cmath> | <cmath> | ||
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</cmath> | </cmath> | ||
| − | There are multiple proofs for the functional equation for Riemann zeta function, and this page presents a | + | == 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 | ||
| + | |||
| + | <cmath> | ||
| + | B_1(x)\triangleq\{x\}-\frac12=-\sum_{n=1}^\infty{\sin(2\pi nx)\over\pi n} | ||
| + | </cmath> | ||
| + | |||
| + | === From <math>\sigma>1</math> to <math>\sigma>-1</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 | ||
| + | |||
| + | <cmath> | ||
| + | \zeta(s)=\frac12+{s\over s-1}-s\int_1^\infty{B_1(x)\over x^{s+1}}\mathrm dx | ||
| + | </cmath> | ||
| + | |||
| + | in which we can extend the ROC of the latter integral to <math>\sigma>-1</math> via repeated integration: | ||
<cmath> | <cmath> | ||
| − | \ | + | \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 |
</cmath> | </cmath> | ||
Revision as of 02:17, 13 January 2021
Introduction
The functional equation for Riemann zeta function is a result due to analytic continuation of Riemann zeta function:
![\[\zeta(s)=2^s\pi^{s-1}\sin\left(\pi s\over2\right)\Gamma(1-s)\zeta(1-s)\]](http://latex.artofproblemsolving.com/7/8/3/78380ea7c8198c2fc2ec42ee52af5523d1e286e9.png)
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
![\[B_1(x)\triangleq\{x\}-\frac12=-\sum_{n=1}^\infty{\sin(2\pi nx)\over\pi n}\]](http://latex.artofproblemsolving.com/a/8/3/a83075739a5adb1f2e8b34b99c53c6f4fe8c7ce5.png)
From 
to 
In this article, we will use the common convention that 
where 
. As a result, we say that the original Dirichlet series definition 
converges only for 
. However, if we were to apply Euler-Maclaurin summation on this definition, we obtain
![\[\zeta(s)=\frac12+{s\over s-1}-s\int_1^\infty{B_1(x)\over x^{s+1}}\mathrm dx\]](http://latex.artofproblemsolving.com/b/d/f/bdf72656f52fee99d364b45f27aeda2e657bad11.png)
in which we can extend the ROC of the latter integral to 
via repeated integration:
![\[\int_1^\infty{B_1(x)\over x^{s+1}}\mathrm dx={B_2(x)\over2x^{s+1}}+{s+1\over2}\int_1^\infty{B_2(x)\over x^{s+2}}\mathrm dx\]](http://latex.artofproblemsolving.com/8/5/c/85c17c07e19d75db2cfb4a79f0d272302bfd0286.png)
