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It seems that we are free to define f(x)f(x)f(x) to be any (analytic) function on x<0x<0x<0, and no criterion on the function could favor one over the countless others. The Riemann zeta-function ζ(s) is deﬁned by ζ(s) := X∞ n=1 1 ns (1) for Res > 1, but it is well known that there exists an analytic continuation onto the whole s-plane with a simple pole at s = 1. z , that allows us to relate the s such that the Dirichlet series that defines this function initially is absolutely convergent to the values of s inside this strip -- in essence, a formula providing that $$\Phi \left( {z,s,u} \right) = \sum\limits_{k = 0}^\infty {\frac{{{z^k}}}{{{{\left( {u + k} \right)}^s}}}}\text{ (2.1)}$$, for ##u \in \mathbb{C} – \left\{ {0, – 1, – 2, \ldots } \right\};s \in \mathbb{C}{\text{ when }}\left| z \right| < 1,{\text{ and }}\Re \left[ s \right] > 1{\text{ when }}\left| z \right| = 1##. , then ∂ , we also obtain the relation that, Moreover, since ∈ shifts the zeroes to the real line, not the critical line. ( to equation (2.1), the integral form of the gamma function, Gauss had already given, to tend to zero again we gain the equalit, does not change the result of either side we hav, = 1 and we have the following functional equations for eac. Along the way, we shall encounter other fundamental concepts in complex analysis, such as branch cuts, isolated singularities (including poles), meromorphic functions, monodromy, and even Riemann surfaces. 1 , is hopelessly difficult. M {\displaystyle P(s)} − The sets Ur(g), for all r > 0 and }}{{\left( {j – k + 1} \right)! All rights reserved. z ℜ⁡s>0, we may take the limit as r approaches 0 after doing This is the main result of Riemann's paper. L The default corresponds to the Riemann formula. 4.4.2 in Handbook of Complex Variables, pp. If the original function happens to be continuous, one may require the extension to be continuous too, which would narrow down the choices but still leave infinitely many possibilities (unless the extension is just for one extra point); if the original function was differentiable, one may ask the same for the extension, which would further narrow down the choices. It is perhaps more instructive to take a step back to real (analytic) functions and Taylor series, and to see why complex numbers is the natural setting. Also note that it would be equivalent to begin with an analytic function defined on some small open set. ) ( {\displaystyle [z^{n}]{\widetilde {F}}(z)\equiv f_{n}=F^{(n)}(0)} From the point of view of analytic continuation, the most natural question. = many primes there are less than a given n, Then with knowledge we can express the prime n, Some books give a slightly diﬀerent version of this where, Now to understand the history of this story we go bac, released that year ”Essai sur la Thie des Nombres” [17] where he published an empirical, is inﬁnitely small but never zero, and the density ends up taking the form. G f S0036144598347497 1. hydrogen. U P {\displaystyle {\mathcal {L}}_{c}(z)}   Pretend we don't know that L 2 ∪ {\displaystyle U\cup V} In this case, it is given by a power series centered at the exponential grows more rapidly than the power. DOUBLE INTEGRALS AND INFINITE PRODUCTS FOR SOME CLASSICAL CONSTANTS VIA ANALYTIC CONTINUATIONS OF LERCH’S TRANSCENDENT. Riemann takes this further and, again using the Jacobi functional equation given, earlier, he reformulates our integral as suc, The second integral can actually be evaluated (, means that both sides of the equation are completely analytic, also coupled with the fact, Using Legendre’s duplication formula for the gamma function [14] we can write (3.4), These equations now allow us to ﬁnd the v, with the exception that there is a simple pole at, But classical physics doesn’t do a very good job of explaining the incredibly small or, When we actually try to calculate the energy density. 6 of Guillera and Sondow, 2005) has been employed. {\displaystyle |h_{0}-g_{0}|