Proper velocity Proper velocity In relativity, proper velocity, also known as celerity, is an alternative to velocity for measuring motion. Whereas velocity relative to an observer is distance per unit time where both distance and time are measured by the observer, proper velocity relative to an observer divides observer-measured distance by the time elapsed on the clocks of the traveling object. Proper velocity equals velocity at low speeds. Proper velocity at high speeds, moreover, retains many of the properties that velocity loses in relativity compared with Newtonian theory. For example, proper velocity equals momentum per unit mass at any speed, and therefore
Proof of the Euler product formula for the Riemann zeta function arithmetic, the partial product when expanded out gives a sum consisting of those terms "n" where "n" is a product of primes less than or equal to "q". The inequality results from the fact that therefore only integers larger than "q" can fail to appear in this expanded out partial product. Since the difference between the partial product and ζ("s") goes to zero when σ > 1, we have convergence in this region. Proof of the Euler product formula for the Riemann zeta function Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis "Variae