# Tag Info

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AFAIK none of proofs is very short and easy. I'll post just a rough sketch (in particular all analytical issues are silently ignored) of a Riemann's original proof based on the Poisson summation formula. Let's define $$\xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s).$$ Riemann's functional equation takes the form $$\xi(s)=\xi(1-s).$$ By definition $$\xi(s)= ... 4 You might try reading some papers in the philosophy of math. They don't require you to do any computations or exercises. I am interested in the philosophy of probability, so the first thing that comes to mind is this paper. The author, Alan Hájek, has many other good papers worth reading. If you want something longer, a classic book is Proofs and ... 2 Ramanujan gave the identity$$e^{\pi \sqrt{163}} = 262537412640768743.99999999999925007... Martin Gardner once claimed as a hoax that $e^{\pi \sqrt{163}}$ is an integer, which apparently confused a lot of people: when they tried to disprove the claim on their calculators, they found that $e^{\pi \sqrt{163}}$ was indeed an integer - but only because their ...

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Felix Klein considered non-Euclidean geometries, like the hyperbolic and spherical geometry. These are the spaces of constant curvature: Euclidean with curvature $0$, hyperbolic with curvature $-1$ and spherical with curvature $1$. The hyperbolic plane is a good toy example, compared to the Euclidean plane. Edit: If you wanted finite geometries, here are ...

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I always am a fan of Dummit and Foote's Abstract Algebra. I also would recommend Ian Stewart's Galois Theory. Another book you could try is David A Cox's Galois Theory. All of these books are more from the algebra side of things and would not have much to say about differential Galois theory. I speak for myself here, but I am not sure there is a book that ...

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I would suggest http://www.ams.org/bookstore-getitem/item=STML-4 http://www.ams.org/bookstore-getitem/item=STML/12 http://www.ams.org/bookstore-getitem/item=STML/21 I am actually solving them now and i feel that they are useful. They have solutions to all problems :P Good Luck!

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1) Evaluation of $I_{\max}(k,n)$ It is clear that $\max x = x$ so $I_{\max}(1,n)=\int\limits_0^1 x_1^n\,dx_1=\frac{1}{n+1}$. Also it is obvious that $\max\limits_{1\le i\le k}x_i=\max\left(x_k,\max\limits_{1\le i\le k-1}x_i\right)$. Then ...

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You might enjoy the answers to this question over on MathOverflow that catalogs some examples of the phenomenon of "eventual counterxamples," particularly the Borwein integrals.

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I picked Chiswell and Hodges "Introduction to Mathematical Logic" which I found in Peter Smith's guide and I personally regard it as one of the best mathematical book I have ever read. It has a lot of exercises many of them have full solutions. I am also self learner and I know how important are exercises with at least answers if not full solutions.

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