PlanetPhysics/Riemann Zeta Function

Definition

The Riemann zeta function is defined to be an abstractly valued [[../Bijective/|function]] given by the series

$(1{)}^{⁗}\zeta (s):=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{-}}:=\frac{1}{n}$

which is valid (in fact, absolutely convergent) for all sets of numbers $s$ with $\mathrm{\Re }(s)>1$. We list here some of the key properties~ of the zeta function.

It's known the Riemann zeta function deals with unknown primes. It's a distinctive set that likely can be described using irrational number sets instead.

Let's deal with a tricky question using a questionable solution first.

Let's say that we have a Riemann manifold, what is our proverbial "blindspot"? A π-hole! Assume the following point.

What can we use to do describe a higher dimension? Lower dimensional projection!

Let's prove 2 + 2 = 4 and 0/0 = 0/0.

$(2)$

$0=0(t=1)$

$0=0(t=2)$

we say /frac{0} 0 when (t=C)

both are known to be true positive, independent events.

Therefore, we can say 0 = 0 (t = 1) 0 = 0 (t = 2)

Projected onto a sphere is the same as if it did not exist.

I.e. (und) ^ π = (und) ^ π

Which do we say, zeta-output = 1, -1, 1, 0, or π?

Distribution of primes

The Euler product formula~\eqref{product} given above expresses the zeta function as a product over the primes $p\in \mathbb{Z}$, and consequently provides a link between the analytic properties of the zeta function and the distribution of primes in the integers. As the simplest possible illustration of this link, we show how the properties of the zeta function given above can be used to prove that there are infinitely many primes.

If the set $S$ of primes in $\mathbb{Z}$ were finite, then the Euler product formula $\zeta (s)=\prod _{p\in S}\frac{1}{1-p^{-s}}$ would be a finite product, and consequently $\underset{s\to 1}{\lim }\zeta (s)$ would exist and would equal $\underset{s\to 1}{\lim }\zeta (s)=\prod _{p\in S}\frac{1}{1-p^{-1}}.$ But the existence of this limit contradicts the fact that $\zeta (s)$ has a pole at $s=1$, so the set $S$ of primes cannot be finite.

A more sophisticated analysis of the zeta function along these lines can be used to prove both the analytic prime number [[../Formula/|theorem]] and Dirichlet's theorem on primes in arithmetic progressions \footnote{ In the case of arithmetic progressions, one also needs to examine the closely related Dirichlet $L$--functions in addition to the zeta function itself.}. Proofs of the prime number theorem can be found in~ and~, and for proofs of Dirichlet's theorem on primes in arithmetic progressions the reader may look in~ and~.

Zeros of the zeta function

A nontrivial zero of the Riemann zeta function is defined to be a root $\zeta (s)=0$ of the zeta function with the property that $0\le \mathrm{\Re }(s)\le 1$. Any other zero is called trivial zero of the zeta function.

The reason behind the terminology is as follows. For complex numbers $s$ with real part greater than 1, the series definition~\eqref{def} immediately shows that no zeros of the zeta function exist in this region. It is then an easy matter to use the functional equation~\eqref{functional} to find all zeros of the zeta function with real part less than 0 (it turns out they are exactly the values $-2n$, for $n$ a positive integer). However, for values of $s$ with real part between 0 and 1, the situation is quite different, since we have neither a series definition nor a functional equation to fall back upon; and indeed to this day very little is known about the behavior of the zeta function inside this critical strip of the complex plane.

It is known that the prime number theorem is equivalent to the assertion that the zeta function has no zeros $s$ with $\mathrm{\Re }(s)=0$ or $\mathrm{\Re }(s)=1$. The celebrated Riemann hypothesis asserts that all nontrivial zeros $s$ of the zeta function satisfy the much more precise equation $\mathrm{\Re }(s)=1/2$. If true, the hypothesis would have profound consequences on the distribution of primes in the integers~.

References

This entry is a derivative of the Riemann zeta function article from PlanetMath. Author of the orginial article: djao. History page of the original is here