# Why do mathematicians care so much about zeta functions?

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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I 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. – mixedmath Jun 18 '12 at 16:19
There 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. – user31373 Jun 18 '12 at 16:43
Have you seen the Riemann Hypothesis (en.wikipedia.org/wiki/Riemann_hypothesis)? This conjecture has major consequences in number theory if true. – Argon Jun 18 '12 at 16:55
@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
@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!) – Matthew Pressland Jun 18 '12 at 17:07

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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However, to this day, no closed form is known for values in the form $\zeta(2n+1)$. This $$\zeta(2n+1)=\frac{(-1)^{n+1}(2\pi)^{2n+1}}{2(2n+1)!} \int_0^1B_{2n+1}(x)\cot(\pi x)\textrm{d}x$$ is a perfectly fine closed form formula for the odd integer values, it's just not as nice (23.2.17 in Abramowitz and Stegun). It's preferable to use vague terms like as nice as opposed to no closed form since it may be more vague but also less jarring. – Peter Sheldrick Jun 18 '12 at 17:29
I'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
@Cam, yes of course $\zeta(3)$ is already a closed form. The term is vague itself. However, such integrals as the one above already are useful for numerics for example. – Peter Sheldrick Jun 18 '12 at 18:06

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.

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