An operator theoretic approach to the Riemann Hypothesis

Abstract

In 2023 an operator theoretic approach to the Prime Number Theorem was introduced by Olsen. In this thesis this approach is examined and applied to give a new, operator theoretic, proof of a different version of the Prime Number Theorem and of the Prime Number Theorem for arithmetic progressions. This approach is then expanded and rearranged to focus on the approximation error in the Prime Number Theorem and then operator theoretic equivalents of the Riemann Hypothesis are derived. Lastly, further properties of the operators involved in these operator theoretic equivalents are discussed.

Popular Abstract

Prime numbers are whole numbers greater than 1 which are divisible only by 1 and itself. The first few prime numbers are 2, 3, 5, 7, and 11. Not only are they very important in mathematics, but they are also widely used in our everyday lives. For example, you have probably surfed the internet securely today using encryption that is based on prime numbers (look up the RSA algorithm if you are interested).

It has been known since the ancient Greeks that there are infinitely many prime numbers (Euclid’s theorem), but as we go further along the number line it gets harder and harder to find them, as they become rarer and more difficult to recognize. Mathematicians in the 19th century have found a formula which tells us approximately where they are, but we have difficulty to this day in determining how accurate this formula is. Of course, we could check its accuracy with a computer for, say, the first several million numbers, but many results in number theory (the area of mathematics where this formula comes from) depend on its accuracy in the long run.

In the 19th century, the work of the great mathematician Bernhard Riemann led to the hypothesis that this formula is as accurate as it can be (still not accurate enough to worry encryption experts though). This is now known as the Riemann Hypothesis and many of the greatest minds of the past 150 years have tried to prove or disprove it. It was included by the famous mathematician David Hilbert in 1900 on his list of 23 important mathematical problems for the 20th century and in 2000 was also named by the Clay Mathematics Institute as one of its 10 Millennium Prize Problems (with a reward of one million dollars for anyone who solves one!). Because this problem is connected to so many results in number theory and has been able to withstand so many attacks for so long and was included on these lists it is considered to be one of the most important unsolved problems in mathematics.

While the formula comes from number theory, this thesis studies its accuracy using a different area of mathematics, called operator theory, and shows how this accuracy is related to certain operators and their properties. Hopefully, a different perspective will shed more light on this accuracy and the Riemann Hypothesis.

• author: van de Scheur, Ramon Arjan
• supervisor: Jan-Fredrik Olsen
• organization: Mathematics (Faculty of Sciences); Centre for Mathematical Sciences
• alternative title: Ett operatorteoretiskt angreppssätt till Riemanns hypotes
• course: MATM03 20232
• year: 2023
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• keywords: Riemann hypothesis, Prime number theorem, Prime number theorem for arithmetic progressions, operator theoretic equivalent, compact operator, prime-counting function, Chebyshev function, Laplace transform, Fourier transform, convolution operator, weak topology, strong topology, compact perturbation, Tauberian theorem, Poisson kernel, holomorphic, self-adjoint, spectrum, trace
• publication/series: Master's Theses in Mathematical Sciences
• report number: LUNFMA-3143-2023
• ISSN: 1404-6342
• other publication id: 2023:E74
• language: English
• additional info: Master’s Theses in Mathematical Sciences 2023:E74; LUNFMA-3143-2023
• id: 9146749
• date added to LUP: 2024-02-01 14:20:37
• date last changed: 2024-02-01 14:20:37