The Riemann zeta function, denoted by $\zeta(s)$ is a function of $s$ and given by
$\zeta(s)=\displaystyle \sum_{n=1}^\infty \frac{1}{n^s}=1+\frac{1}{2^s}+\frac{1}{3^s}+\cdots$,
where $s$ is a complex variable and the function is absolutely convergent when the real value of s is greater than 1.
Riemann zeta function can also be expressed in terms of integral as
$\zeta(s)=\frac{1}{\Gamma(s)}\int_0^\infty \frac{x^{s-1}}{e^x-1}dx$
This is the convergent series for any $s > 1$.Consider the case when $s=2$, then
$\zeta(2)=\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}=1+\frac{1}{4}+\frac{1}{9}+\cdots = \frac{\pi^2}{6}$.
For $s=1$, it is divergent infinite series, called the harmonic series as given by
$\zeta(1)=\displaystyle \sum_{n=1}^\infty \frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots=\infty$.
For even positive integers, the Riemann zeta function can be expressed in terms of the Bernoulli number ($B_0=1, B_1=\pm\frac{1}{2}, B_2=\frac{1}{6}, B_3=0, B_4=\frac{-1}{30}, B_5=0, B_6=\frac{1}{42}, B_7=0, B_8=\frac{-1}{30}, B_9=0,\\ B_{10}=\frac{5}{66}, \cdots $ and so on) because all odd Bernoulli's number are zero except 1. More about Bernoulli's number are referred to https://en.wikipedia.org/wiki/Bernoulli_number . The expression is given by
$\zeta(2n)= \frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}$ for any $n \ge 1$.
For negative integers, Riemann zeta function is also expressed in terms of Bernoulli number as
$\zeta(-n)=-\frac{B_{n+1}}{n+1}$ .
Hence $\zeta(-1)=1+2+3+4+\cdots= -\frac{B_2}{2}=-\frac{1}{12}$. This also explains the Grandi's series as we discussed earlier. In this case, the series is divergent and this specific infinite sum of natural numbers have been used to explain the string theory. More about this is available in the following link https://plus.maths.org/content/infinity-or-just-112.
$\zeta(s)=\displaystyle \sum_{n=1}^\infty \frac{1}{n^s}=1+\frac{1}{2^s}+\frac{1}{3^s}+\cdots$,
where $s$ is a complex variable and the function is absolutely convergent when the real value of s is greater than 1.
Riemann zeta function can also be expressed in terms of integral as
$\zeta(s)=\frac{1}{\Gamma(s)}\int_0^\infty \frac{x^{s-1}}{e^x-1}dx$
This is the convergent series for any $s > 1$.Consider the case when $s=2$, then
$\zeta(2)=\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}=1+\frac{1}{4}+\frac{1}{9}+\cdots = \frac{\pi^2}{6}$.
For $s=1$, it is divergent infinite series, called the harmonic series as given by
$\zeta(1)=\displaystyle \sum_{n=1}^\infty \frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots=\infty$.
For even positive integers, the Riemann zeta function can be expressed in terms of the Bernoulli number ($B_0=1, B_1=\pm\frac{1}{2}, B_2=\frac{1}{6}, B_3=0, B_4=\frac{-1}{30}, B_5=0, B_6=\frac{1}{42}, B_7=0, B_8=\frac{-1}{30}, B_9=0,\\ B_{10}=\frac{5}{66}, \cdots $ and so on) because all odd Bernoulli's number are zero except 1. More about Bernoulli's number are referred to https://en.wikipedia.org/wiki/Bernoulli_number . The expression is given by
$\zeta(2n)= \frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}$ for any $n \ge 1$.
For negative integers, Riemann zeta function is also expressed in terms of Bernoulli number as
$\zeta(-n)=-\frac{B_{n+1}}{n+1}$ .
Hence $\zeta(-1)=1+2+3+4+\cdots= -\frac{B_2}{2}=-\frac{1}{12}$. This also explains the Grandi's series as we discussed earlier. In this case, the series is divergent and this specific infinite sum of natural numbers have been used to explain the string theory. More about this is available in the following link https://plus.maths.org/content/infinity-or-just-112.
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