**Background:**

The Riemann Hypothesis is a conjecture that states that the nontrivial zeros of the Riemann zeta function all have real part 1/2. There are widespread applications in pure mathematics of the Riemann Hypothesis, including the analysis of the distribution of prime numbers. The Riemann zeta function itself is a function, denoted ζ(

*s*), in which

*s*can be any complex number not equal to 1. In turn, all values ζ(

*s*) are complex. The function has two types of zeros: trivial zeros, located at the even integers; and nontrivial zeros, whose locations are currently unknown. The basis of solving this problem is in determining where these nontrivial zeros are located.

**Your Task:**

The Hypothesis states that the nontrivial zeros of ζ(

*s*) all have real part 1/2. Thus they all are of the form 1/2+

*ti,*where

*t*is a real number. Using calculus, complex analysis, and perhaps some number theory, try to make headway into this problem. In doing so, investigate the nontrivial zeros as well as the application to prime number distribution. This Millennium Prize problem comes with a $1,000,000 cash prize.