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The Riemann hypothesis is a conjecture about the distribution of prime numbers. It states that all nontrivial zeros of the Riemann zeta function have a real part equal to $\frac{1}{2}$. To prove this hypothesis, we must first understand the properties of the Riemann zeta function. The Riemann zeta function is defined as:

where s is a complex number. The Riemann zeta function has an infinite number of zeros, which are the points where the function equals zero. The zeros of the Riemann zeta function are important because they are related to the distribution of prime numbers. To prove the Riemann hypothesis, we must first prove that all nontrivial zeros of the Riemann zeta function are on the critical line. This line is the imaginary axis of the complex plane with a real part of $\frac{1}{2}$. In other words, all nontrivial zeros of the Riemann zeta function have a real part equal to $\frac{1}{2}$. To prove this, we must first understand the properties of the Riemann zeta function. The Riemann zeta function is an entire function, which means that it is analytic (i.e. differentiable) everywhere in the complex plane. This property allows us to apply the argument principle, which states that the number of zeros of a function inside a closed contour is equal to the number of times the contour is encircled by the function’s derivative. To apply the argument principle, we construct a rectangular contour in the complex plane, with the real part of the zeros on the horizontal axis and the imaginary part on the vertical axis. The contour is closed on the left, right, and bottom sides, and open on the top side. We then apply the argument principle to the Riemann zeta function. Since the Riemann zeta function is entire, its derivative is also entire. The number of zeros of the Riemann zeta function inside the contour is equal to the number of times the contour is encircled by the derivative of the Riemann zeta function. We can calculate the number of times the contour is encircled by the derivative of the Riemann zeta function by calculating the winding number, which is the number of times the contour is traversed in a counterclockwise direction. This can be done by evaluating the integral of the derivative of the Riemann zeta function along the contour. The derivative of the Riemann zeta function is: