Why is it that so many people care so much about zeta functions? Why do people write books and books specifically about the theory of Riemann Zeta functions?
What is its purpose? Is it just to develop small areas of pure mathematics?
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6I have completely and totally rewritten your question to make it more pleasant to the eye. Please review it and make sure that I captured your question. – davidlowryduda Jun 18 '12 at 16:19
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@mixedmath - Thank you so much – Victor Jun 18 '12 at 16:20
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1I've removed "ivory tower". – ShreevatsaR Jun 18 '12 at 16:30
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@ShreevatsaR: Ok - grand. – davidlowryduda Jun 18 '12 at 16:38
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@Shree What was wrong with "ivory tower"? – Adrián Barquero Jun 18 '12 at 16:39
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17There are very few people who care about Riemann Zeta function. Far fewer than the number of people who care about Nicki Minaj. Maybe you should address the question "why? what is the purpose?" to the latter group. – Jun 18 '12 at 16:43
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4Have you seen the Riemann Hypothesis (http://en.wikipedia.org/wiki/Riemann_hypothesis)? This conjecture has major consequences in number theory if true. – Argon Jun 18 '12 at 16:55
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3@AdriánBarquero: Nothing wrong with "ivory tower", just that the original post did not contain it and IMHO we should be extra cautious when rephrasing questions. (Also, someone before me had left a comment with 2 or 3 upvotes pointing out this new addition, so I thought it simplest to just remove it.) Also, its removal seems to have prompted the OP to clarify the intent better. BTW, I thank mixedmath for editing the question into a more readable form. – ShreevatsaR Jun 18 '12 at 17:05
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6@AdriánBarquero I would add that at least in British English, "ivory tower" carries a negative connotation that was probably not intended. (Even if it was intended, it's politer not to mention it!) – mdp Jun 18 '12 at 17:07
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1Purpose? To have fun. (On a more general note, I believe the ultimate purpose of everything is to attain bliss.) – Jun 18 '12 at 17:22
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2In this article http://arxiv.org/abs/1101.3116v1 authors present brief introduction to Riemann Zeta function and Riemann Hypothesis. Moving forward there are several examples of Riemann Hypothesis applications in physics, which of course is closely connected to Zeta function. – qoqosz Jun 18 '12 at 17:36
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@MattPressland Oh I didn't know that. That's good to know. Thanks for clarifying. I hadn't noticed that "ivory tower" wasn't in the original post. – Adrián Barquero Jun 18 '12 at 17:43
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2Ivory Tower: From the 19th century it has been used to designate a world or atmosphere where intellectuals engage in pursuits that are disconnected from the practical concerns of everyday life. As such, it usually carries pejorative connotations of a wilful disconnect from the everyday world; esoteric, over-specialized, or even useless research; and academic elitism, if not outright condescension. In American English usage it is a shorthand for academia or the university, particularly departments of the humanities. – Pedro Jun 18 '12 at 20:20
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3Related question. The short answer, in my mind: the theory surrounding the Riemann zeta function has expanded throughout number theory and mathematics to the point it touches so many areas and yet still hasn't been fully penetrated with so many angles to it. This gives the reasonable impression that the $\zeta$ and related objects lie at the heart of quite a lot of number theory as we currently know it, and is very deep indeed. – anon Jun 18 '12 at 20:24
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For one thing, the Riemann Zeta function has many interesting properties. No one knew of a closed form of $\zeta (2)$ until Euler famously found it, along with all the even positive integers: $$\zeta(2n) = (-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}$$
However, to this day, no nice closed form is known for values in the form $\zeta(2n+1)$.
Another major need of the Zeta function is relating to the Riemann hypothesis. This conjecture if fairly simple to understand. It essentially hypothesizes that the nontrivial zeros of the zeta function have a real part of 1/2. This hypothesis, if proven true, has major implications in number theory and the distribution of primes.
The Riemann zeta function also occurs in many fields and appears occasionally when evaluating different equations, just as many other functions do.
Lastly, the sum
$$\sum_{n=1}^{\infty} \frac{1}{n^s}$$
is a very natural one to try and study and evaluate and is especially interesting because of the above-mentioned properties and more.
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7I'm not sure I agree. After all, an integral is just short-hand for a limit of sums. In any case, I don't think it's a particularly well-defined concept, as there's always an implied "closed, relative to blah" that goes unspecified. For instance, if I were to simplify a complicated expression to, e.g., $2\pi\zeta(3)$, I might call that a closed form for the initial expression! Yet I think it would certainly be a stretch to call the notation "$\zeta(3)$" a closed form for itself. – Cam McLeman Jun 18 '12 at 17:57
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People write books about the theory of Riemann Zeta functions because there is sufficiently developed theory and enough applications to warrant a dedicated book, much the same way that people write books specifically about elliptic curves or Schrodinger's equation.
As for the research interest in the Riemann Hypothesis, this MO thread gathers some of its consequences and gives an idea of to which parts of mathematics it can apply.
And as for more "popular" interest, here's a quote from one of the aforementioned books, Edwards' Riemann's Zeta Function:
The experience of Riemann's successors with the Riemann hypothesis has been the same as Riemann's -- they also consider its truth "very likely" and they also have been unable to prove it. ... the attempt to solve this problem has occupied the best efforts of many of the best mathematicians of the twentieth century. It is now unquestionably the most celebrated problem in mathematics and it continues to attract the attention of the best mathematicians, not only because it has gone unsolved for so long but also because it appears tantalizingly vulnerable and because its solution would probably bring light to new techniques of far-reaching importance.
That's from 1974, and is probably even more applicable today.
