How Riemann found his hypothesis?

Question

We know that according Riemann hypothesis all non trivial zeros of dzeta function lie on (0.5, x) line on complex surface. I wonder how Reieman found that idea. Does he just found first few zeros by brute force method and since they all lie on 1/2 line, he says that probably all of them has this property? Or there are deeper reasons to say that? Or other third possibility: Riemann was just a genius and nobody has idea how he found his hypothesis?

Answer 1

Well, I happened to read Riemann's 1859 paper just yesterday out of curiosity, so here's my brief answer (warning: I know very little about the history, and even less about the progress. All I can offer you here is a surface-level answer). In the paper, he defines his famous zeta function $\zeta(s)$ and talks about prime numbers and so on. As a brief description of the first few (3-4) pages, here's what happens:

• He defines $\zeta(s)$ for $\text{Re}(s)>1$ using the familiar series definition $\sum_{n=1}^{\infty}\frac{1}{n^s}$. The reason he's interested in this function is that it is very nicely related to prime numbers, namely $\zeta(s)=\prod(1-p^{-s})^{-1}$, the product taken over all positive primes (Euler's formula).

• He talks about relationship to the Gamma function $\Gamma(s)$ through the identity $\Gamma(s)\zeta(s)=\int_0^{\infty}\frac{x^{s-1}}{e^x-1}\,dx$ (though in the paper, the notation $\Pi(s-1)$ is used instead of $\Gamma(s)$).

• Next, he immediately goes on to talk about analytic continuation of $\zeta(s)$ be deforming the contour of integration; using Jacobi's theta identity blablabla.

While doing the analytic continuation, he comes across an expression and gives it a name. In the paper (page 3), he defines $s=\frac{1}{2}+it$ and he defines the function \begin{align} \xi(t)&:=\Pi\left(\frac{s}{2}\right)(s-1)\pi^{-s/2}\zeta(s)\\ &:=\Gamma\left(\frac{s}{2}+1\right)(s-1)\pi^{-s/2}\zeta(s)\\ &=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s) \end{align} This was Riemann's original $\xi(t)$ function. Nowadays this would be denoted at $\Xi(t)$, while expression on the right is what we define as $\xi(s)$, so Riemann's original function is actually $\xi_{\text{new}}\left(\frac{1}{2}+is\right)$. But for the rest of this, I shall stick to Riemann's notation. This is an entire function of $t$ (holomorphic on $\Bbb{C}$), and he now talks about zeros of this function, which I quote:

The number of roots of $\xi(t)=0$ whose real parts lie between $0$ and $T$ is approximately \begin{align} =\frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}; \end{align} because the integral $\int d\log\xi(t)$, taken in a positive sense around the region consisting of the values of $t$ whose imaginary parts lie between $\frac{1}{2}i$ and $-\frac{1}{2}i$ and whose real parts lie between $0$ and $T$, is (up to a fraction of the order of magnitude of the quantity $\frac{1}{T}$) equal to $\left(T\log\frac{T}{2\pi}-T\right)i$; this integral however is equal to the number of roots of $\xi(t)=0$ lying within in this region, multiplied by $2\pi i$. One now finds indeed approximately this number of real roots within these limits, and it is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation.

The last bold is mine. The assertion that $\xi(t)=0$ having real roots is the same as the roots of $\zeta(s)$ lying on the critical line $\text{Re}(s)=\frac{1}{2}$ (recall Riemann set $s=\frac{1}{2}+it$), which is precisely what we today call Riemann's hypothesis. So, Riemann seemed to be considering zeros of his $\xi(t)$ function, and got some estimates on how many zeros lie in a certain region, and hypothesized that all the roots must in fact be real.