Riemann Hypothesis
A complex function theory exploration
J. van Leeuwen (TU Delft - Electrical Engineering, Mathematics and Computer Science)
W.G.M. Groenevelt – Mentor (TU Delft - Electrical Engineering, Mathematics and Computer Science)
J.A.M. de Groot – Graduation committee member (TU Delft - Electrical Engineering, Mathematics and Computer Science)
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Abstract
The Riemann zeta function, ζ(s) = Σ n-s, was initially only studied for real s>1. It is not difficult to see this can be extended to complex valued s provided the real part of s is greater than 1. We will study the zeta function on this domain first, and find the Euler product, Π 1/(1-p-s), where p is prime, which links the zeta function to the primes. To do so, we will study the theory of infinite products.
Next we wish to find an analytic continuation of ζ(s) to the entire complex plane, and we will do so using the Euler-Maclaurin summation formula. To prove the formula holds we will study Bernoulli numbers and polynomials, which also allows us to find specific values of the zeta function. Using the analytic continuation we will then obtain the functional equation, which allows us to study the function in more detail. The functional equation gives us the trivial zeroes, at which point we are able to study the critical strip 0<Re(s)<1, the critical line s=½+it, and attempt to find the non-trivial zeroes.
We investigate the Riemann hypothesis and prove certain classical results, such as Hardy's theorem, which states there are infinitely many zeroes on the critical line. Finally, we discuss certain related theorems from the literature and end by a brief discussion of the consequences of the Riemann hypothesis, as well as what the Generalised Riemann Hypothesis states.