This extra credit is an individual project, and is based on the Reimann Hypothesis.

In mathematics, the Reimann Hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $0\mathrm{.}5.$ Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann $(1859),$ after whom it is named.

The expansion of the Reimann zeta function can be seen by the following convergent infinite series:

$(s)={\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^s}$

$[50$ pts$]$ Evaluate this expression for a range of $x$ between $-30$ and $30.$ The value for $s$ shall be equal to $0\mathrm{.}5+$ ix

$1\mathrm{.}1$ Create a range of $x$ with an increment of your choosing between $-30$ and $30$

$1\mathrm{.}2$ Create the range for s where $s=0\mathrm{.}5+ix$

$1\mathrm{.}3$ Solve the expansion of the Reimann zeta function for all values of $s$ using a combination of for and while loops

$1\mathrm{.}4$ Matlab also has a function for this, called "zeta". Use this built$-$in Matlab function to evaluate the function over the same range. Note: the results will be closer but likely not exactly the same

$1\mathrm{.}5$ Plot both functions over the same figure. Note: Because these are complex numbers, there are real and imaginary portions of both results. The plot should have $4$ lines.

Real portion of your result in a solid blue line

Imaginary portion of your result in a solid red line

Real portion from the built$-$in zeta function in dashed blue line

Imaginary portion from the built$-$in zeta function in dashed red line

$1\mathrm{.}6$ Plots must have titles and Legends with the grid on