Euler/Riemann Point of Departure in Riemann's 1859 paper containing RH

• MHB
• Greg
In summary, the Euler/Riemann Point of Departure is a concept introduced by mathematicians Leonhard Euler and Bernhard Riemann in their study of the Riemann zeta function. It is a point on the complex plane where the zeta function has a singularity, and its location is closely related to the Riemann Hypothesis. Riemann's 1859 paper proposed a new approach to the distribution of prime numbers using complex analysis and introduced the Riemann zeta function. The Riemann Hypothesis, proposed in the same paper, states that all non-trivial zeros of the zeta function lie on the critical line. Despite significant progress, the RH remains unsolved and is considered one
Greg
Gold Member
MHB
In his 1859 paper entitled "On the Number of Primes Less than a Given Magnitude", Riemann gives as his point of departure the equation

$$\displaystyle \prod\frac{1}{1-\frac{1}{p^s}}=\sum\frac{1}{n^s}$$

where $p$ is all primes and $n$ is all natural numbers. The function of the complex variable $s$, wherever these expressions converge, is called by Riemann $\zeta(s)$.

Any thoughts on how to prove this equation?

It is called the Euler product formula for the Riemann zeta function.

Wiki gives 2 proofs here.

What is the Euler/Riemann Point of Departure in Riemann's 1859 paper containing RH?

The Euler/Riemann Point of Departure is a concept introduced by Bernhard Riemann in his 1859 paper "On the Number of Primes Less Than a Given Magnitude". It refers to the starting point for Riemann's investigations into the distribution of prime numbers, which ultimately led to the formulation of the Riemann Hypothesis (RH).

Why is the Euler/Riemann Point of Departure important in relation to RH?

The Euler/Riemann Point of Departure is important because it marks the beginning of Riemann's groundbreaking work on prime numbers and the Riemann Hypothesis. It provides a starting point for understanding the distribution of prime numbers and serves as the foundation for Riemann's subsequent investigations.

What does the Euler/Riemann Point of Departure tell us about prime numbers?

The Euler/Riemann Point of Departure tells us that the distribution of prime numbers is closely related to the behavior of the Riemann zeta function. This function, originally studied by Leonhard Euler, plays a crucial role in Riemann's formulation of the Riemann Hypothesis.

How does the Euler/Riemann Point of Departure relate to the Riemann zeta function?

The Euler/Riemann Point of Departure provides a starting point for understanding the behavior of the Riemann zeta function. It allows us to make connections between the distribution of prime numbers and the properties of the zeta function, which is a key component of the Riemann Hypothesis.

What impact has the Euler/Riemann Point of Departure had on mathematics?

The Euler/Riemann Point of Departure has had a significant impact on mathematics, particularly in the field of number theory. It paved the way for Riemann's groundbreaking work on the Riemann Hypothesis and has inspired countless mathematicians to continue investigating this important problem in mathematics.

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