The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation

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The Riemann zeta function $latex {zeta(s)}&fg=000000$ is defined in the region $latex {hbox{Re}(s)>1}&fg=000000$ by the absolutely convergent series

$latex displaystyle zeta(s) = sum_{n=1}^infty frac{1}{n^s} = 1 + frac{1}{2^s} + frac{1}{3^s} + ldots. (1)&fg=000000$

Thus, for instance, it is known that $latex {zeta(2)=pi^2/6}&fg=000000$, and thus

$latex displaystyle sum_{n=1}^infty frac{1}{n^2} = 1 + frac{1}{4} + frac{1}{9} + ldots = frac{pi^2}{6}. (2)&fg=000000$

For $latex {hbox{Re}(s) leq 1}&fg=000000$, the series on the right-hand side of (1) is no longer absolutely convergent, or even conditionally convergent. Nevertheless, the $latex {zeta}&fg=000000$ function can be extended to this region (with a pole at $latex {s=1}&fg=000000$) by analytic continuation. For instance, it can be shown that after analytic continuation, one has $latex {zeta(0) = -1/2}&fg=000000$, $latex {zeta(-1) = -1/12}&fg=000000$, and $latex {zeta(-2)=0}&fg=000000$, and more generally

$latex displaystyle zeta(-s) = – frac{B_{s+1}}{s+1} (3)&fg=000000$

for $latex {s=1,2,ldots}&fg=000000$, where $latex {B_n}&fg=000000$ are the Bernoulli numbers. If one formally applies

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Reaction Diffusion Systems/Phenomena

Taeuber U.C.-Critical Dynamics_ A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior-CUP (2014)

Generic reaction-diffusion models are in fact utilized to describe a multitude of phenomena in various disciplines, ranging from population dynamics in ecology, competition of bacterial colonies in microbiology, dynamics of magnetic monopoles in the early Universe
in cosmology, equity trading on the stock market in economy, opinion exchange
in sociology, etc. More concrete physical applications systems encompass excitons kinetics in organic semconductors, domain wall interactions in magnets, and
interface dynamics in growth models. Yet most of our current knowledge in this
area stems from extensive computer simulations, and actual experimental realizations allowing accurate quantitative analysis are still deplorably rare