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In Arfken's Mathematical Methods for Physicists, there is a subsection of the "Infinite Series" chapter which covers the Bernoulli numbers, Euler-Mascheroni integration (or summation), and the connection these have with the Riemann zeta function. However, apart from a few nifty math problems these solve (explicit expressions for some sums and integrals), I can't see the use of these concepts in physics. There were a few problems at the end of the chapter that said that some practice integrals showed up in QED corrections, so that's a start. I would like to know where else these ideas show up in physics, if they do.

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closed as too broad by Kyle Kanos, Sebastian Riese, tpg2114, CuriousOne, Qmechanic Mar 7 '16 at 22:56

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I do not remember ever seeing Bernoulli numbers in physics, but I have seen Riemann zeta function pop out in quantum statistical physics.

The integrals of the type

$$g_n(z)=\frac{1}{\Gamma (n)}\int _0^{+\infty }dx\frac{x^{n-1}}{z^{-1}e^x-1}\tag{1} $$

often occur in calculations involving ideal Bose gases. For example, at low temperature, the concentration of a Bose gas is proportional to $g_{3/2}(z)$. For $z=1$, $(1)$ reduces to

$$g_n(1)=\zeta (n)$$

For further information you can read Tong's lectures on QSM.
For a general value of $z$, $g_n(z)$ can be expressed as

$$g_n(z)=\sum _{k=1}^{\infty }\frac{z^k}{k^n}$$

which is actually a polylogarithm, which, for some specific values of its parameters can related to Bernoulli polynomials $B_n(x)$. Bernoulli numbers are defined as $B_{n}=B_{n}(0)$, so they may possibly occur in physics.

On another note, Riemann zeta function may occur in electrostatics. I remember that the electric potential of a finite cylinder (I forgot the boundary conditions) was proportional to $\zeta (3)$, also known as Apéry's constant, which also appears in other areas of physics.

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In quantum field theory, especially in the treatment of divergent series and divergent integrals (like Feynman integrals which arise from calculation of self energies for example) and during the process called "regularization", lots of "Euler Mascheroni" constants arise.

For example, I remember during the calculation of the Photon self-energy (vacuum polarization) in a $2\omega$ dimensiona spacetime is provided by the parametric integral:

$$\text{reg}\Pi(k^2, M^2) = (-2)\frac{\alpha}{\pi}\int_0^1 x(1-x) \left\{\frac{1}{\epsilon} - \gamma -\ln\left[\frac{M^2 - x(1-x)k^2}{2\pi \mu^2}\right]+ \mathcal{O}(\epsilon)\right\}\ \text{d}x$$

in which you see the Euler Mascheroni $\gamma$ constant arising in the integration.

Riemann Zeta Function arises instead when you are trying to regularize divergent series. I bet at least once you saw this:

$$\zeta(0) = -\frac{1}{2}$$

A regularization of the infinite series

$$\zeta(0) = \sum_{k = 1}^{+\infty} \frac{1}{k^0} = 1 + 1 + 1 + 1 + \cdots $$

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These things will show up somewhat generically whenever you needs to use infinite series in physics. So for instance, as your book mentioned this happens a lot of in quantum field theory. However, another big area of physics where you get this is condensed matter physics where you want too count up all the modes in a material and their contributions to some effect. It's worth noting that the point about condensed matter is somewhat of a cheat since there are some very deep connections between condensed matter physics and quantum field theory, in fact we use quantum field theory to model a lot of condensed matter systems.

These infinite sums also show up in electromagnetism for instance when you want to describe say the potential between two plates held at certain potentials you use Fourier series to describe the potential, in which case you can get all sorts of interesting infinite series with these properties.

These are just a couple examples but physicists describing things using infinite series because it's easy to approximate and you can just choose the accuracy of your approximation. You should look up perturbation theory if you're interested. I'm sure lots of people will show up with lots more examples of where you can use infinite series in various areas of physics.

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