Use ramanujan series to calculate pi8/14/2023 Watching the video is like watching a magician and trying to spot them slipping the rabbit into the hat. So how did the people in the Numberphile video "prove" that the natural numbers all add up to -1/12? The real answer is that they didn’t. This is one way of making sense of Ramanujan’s mysterious expression. If you now make the mistake of believing that for, then you get the (wrong) expression What value do you get when you plug into the zeta function? You’ve guessed it: When you plug in values, the zeta function gives you a finite output. So now we have a function that agrees with Euler’s zeta function when you plug in values. (Making this new function give you finite values for involves cleverly subtracting another divergent sum, so that the infinity from the first divergent sum minus the infinity from the second divergent sum gives you something finite.) This method of extension is called analytic continuation and the new function you get is called the Riemann zeta function, after the 19th cenury mathematician Bernhard Riemann. In other words, there is a way of defining a new function, call it so that forĪnd for the function has well-defined, finite values. Using some high-powered mathematics (known as complex analysis, see the box) there is a way of extending the definition of the Euler zeta function to numbers less than or equal to 1 in a way that gives you finite values. That's the Riemann zeta function.īut there is also another thing you can do. Since real numbers are also complex numbers, we can regard it as a complex function and then apply analytic continuation to get a new function, defined on the whole plane but agreeing with the Euler zeta function for real numbers greater than 1. The Euler zeta function is defined for real numbers greater than 1. This method of extending the definition of a function is known as analytic continuation. Some set of inputs, then (up to some technical details) you can know the value of the function everywhere else on the complex plane. One amazing thing about functions of complex numbers is that if you know the function sufficiently well for Just as you can define functions that take real numbers as input you can define functions that take complex numbers as input. And while the real numbers correspond to all the points along an infinitely long line, the complex numbers correspond to all the points on a plane, containing the real number line. The real numbers are part of a larger family of numbers called the complex numbers. The same is true for any other values of less than or equal to 1: the sum diverges.Īs it stands the Euler zeta function S(x) is defined for real numbers x that are greater than 1. So you recover our original sum, which, as we know, diverges. But what happens when you plug in a value of that is less than 1? For example, what if you plug in ? Let’s see. is what’s called a function, and it’s called the Euler zeta function after the prolific 18th century mathematician Leonhard Euler. For every, the expression has a well-defined, finite value. Now what happens when instead of raising those natural numbers in the denominator to the power of 2, you raise it to some other power ? It turns out that the corresponding sumĬonverges to a finite value as long as the power is a number greater than. Then the results you get get arbitrarily close, without ever exceeding, the number Mathematicians say the sum converges to, or more loosely, that it equals If you take the sequence of partial sums as we did above, You might recognise this as the sum you get when you take each natural number, square it, and then take the reciprocal: To understand what that is, first consider the He had been working on what is called theĮuler zeta function. But Ramanujan knew what he was doing and had a reason for In the work of the famous Indian mathematician Srinivasa Ramanujan in 1913 So where does the -1/12 come from? The wrong result
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