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

  • Thread starter Thread starter Greg
  • Start date Start date
  • Tags Tags
    Paper Point
AI Thread Summary
Riemann's 1859 paper introduces the Euler product formula for the Riemann zeta function, expressed as the equation linking prime numbers and natural numbers. This formula highlights the relationship between the distribution of prime numbers and the zeta function, denoted as $\zeta(s)$. The discussion seeks insights on proving this equation, emphasizing its significance in number theory and the Riemann Hypothesis (RH). Participants reference existing proofs available on Wikipedia, indicating a collaborative effort to deepen understanding. The exploration of this foundational equation remains crucial for advancing research in prime number theory and the RH.
Greg
Gold Member
MHB
Messages
1,377
Reaction score
0
In his 1859 paper entitled "On the Number of Primes Less than a Given Magnitude", Riemann gives as his point of departure the equation

$$\prod\frac{1}{1-\frac{1}{p^s}}=\sum\frac{1}{n^s}$$[/size]

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?

All comments welcome. :)
 
Mathematics news on Phys.org
It is called the Euler product formula for the Riemann zeta function.

Wiki gives 2 proofs here.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
Back
Top