Proof of Inf. Riemann Zeta Function Zeros at re(s)=1/2

In summary, the Riemann Hypothesis is an unsolved problem in mathematics that proposes a pattern for the distribution of prime numbers. It is closely related to the Riemann zeta function, a mathematical function that is defined as the sum of the reciprocals of all positive integers raised to a given power. Proving the Riemann Hypothesis has significant implications in mathematics, cryptography, and physics, but as of now, it remains unsolved. The Riemann zeta function also has connections to various branches of mathematics and has applications in physics, particularly in quantum mechanics and string theory.
  • #1
epkid08
264
1
Does anybody know where I can find the proof that an infinite number of zeros of the riemann zeta function exist when re(s) = 1/2?
 
Physics news on Phys.org
  • #2
Selberg, A. "On the Zeros of Riemann's Zeta-Function." Skr. Norske Vid.-Akad. Oslo, No. 10, 1942.
 

1. What is the Riemann Hypothesis?

The Riemann Hypothesis is one of the most famous unsolved problems in mathematics. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line, re(s)=1/2. In simpler terms, it proposes a pattern for the distribution of prime numbers.

2. What is the Riemann zeta function?

The Riemann zeta function is a mathematical function that plays a crucial role in number theory. It is defined as the sum of the reciprocals of all positive integers raised to a given power. The function is denoted by ζ(s) and is defined for complex values of s.

3. Why is proving the Riemann Hypothesis important?

The Riemann Hypothesis has far-reaching implications in mathematics and other fields. If proven true, it would provide a deeper understanding of the distribution of prime numbers and could potentially lead to new techniques for solving other mathematical problems. It also has applications in cryptography and physics.

4. What is the current state of the proof of the Riemann Hypothesis?

As of now, the Riemann Hypothesis remains unsolved. Many mathematicians have attempted to prove it, and there have been some partial results, but a complete proof has not yet been found. The Riemann zeta function and its properties continue to be studied extensively by mathematicians around the world.

5. How does the Riemann zeta function relate to other branches of mathematics?

The Riemann zeta function has connections to many areas of mathematics, including number theory, complex analysis, and algebraic geometry. It also has applications in physics, particularly in the study of quantum mechanics and string theory. Its properties and the Riemann Hypothesis also have implications in the study of random matrices and the distribution of eigenvalues.

Similar threads

  • Topology and Analysis
Replies
3
Views
1K
  • General Math
Replies
4
Views
981
Replies
5
Views
3K
  • Linear and Abstract Algebra
Replies
10
Views
4K
Replies
2
Views
1K
Replies
2
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
2K
Replies
6
Views
1K
Replies
11
Views
2K
  • Topology and Analysis
Replies
17
Views
2K
Back
Top