- #1
overlook1977
- 11
- 0
This may be a stupid question, but I do not understand why the Euler sum is infinite for zeta=1. Why is "1+1/2+1/3+1/4+... " infinite, but zeta=2 (1+1/4+1/9+...) not?
mathman said:1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+...>
1+1/2+1/4+1/4+1/8+1/8+1/8+1/8+...=
1+1/2+1/2+ 1/2+... which diverges.
The Euler Sum, also known as the Euler-Mascheroni Constant, is defined as the limit of the difference between the harmonic series and the natural logarithm function. When the Riemann Zeta function, represented by ζ, is evaluated at 1, it gives a value of infinity for the Euler Sum. This means that the sum of the harmonic series diverges or goes to infinity when the Riemann Zeta function is evaluated at 1.
The Riemann Zeta function is a mathematical function that is closely related to the distribution of prime numbers. It is defined as ζ(s) = ∑1/n^s, where s is a complex number with a real part greater than 1. It is a famous function in number theory and has many interesting properties and applications.
The Euler Sum only diverges or goes to infinity when the Riemann Zeta function is evaluated at 1 because of the properties of the harmonic series. When the Riemann Zeta function is evaluated at other values, it converges or has a finite value. This is because the harmonic series is convergent for all s values greater than 1 except for 1 itself.
The fact that the Euler Sum is infinite for Zeta=1 has significant implications in mathematics, specifically in number theory and analysis. It is closely related to the distribution of prime numbers and has connections to other famous mathematical constants, such as the Golden Ratio and the Gamma function. The infinite value of the Euler Sum also has applications in physics, particularly in quantum field theory.
Yes, there is a mathematical proof for the Euler Sum being infinite for Zeta=1. It involves using techniques from number theory and complex analysis, such as the Euler-Maclaurin summation formula and analytic continuation. The proof is quite complex and requires a strong background in mathematics to understand.