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in this video we are going to be exploring the raymond zeta function mathematicians have been interested in this function for hundreds of years not only does this function have outputs that we could talk forever about but it also has applications in physics and encodes a lot of information concerning the location of prime numbers the zeta function is the summation of the reciprocal of the natural numbers raised to some complex exponent s this might sound a bit confusing for example if our input was s is equal to 2 the function would be the summation of 1 over 1 squared plus one over two squared plus one over three squared and so on all the way to infinity this function was originally only defined for when s had a real part greater than one however through a method called analytical continuation mathematicians have been able to expand the zeta function's domain for all complex values of s while this is another topic that definitely deserves its own video just think of this as a method used to expand the domain of a function back in 1650 a mathematician named pietro mongoli posed a problem he wanted to know the exact value of zeta of two in other words what is the exact sum of the reciprocal of the squares of all natural numbers this was called the basil problem the solution eluded some of the greatest mathematicians at the time such as the bernoulli family leibniz and john wallace it was not until 1736 when leonard euler solved the problem giving him instant fame and recognition in the math community so how did he solve the problem let's solve this problem the way that euler did firstly let's start with the maclaurin series for sine of x then for reasons that will become apparent later let's divide both sides by x secondly let's try to write sine x as the product of its roots starting with pi minus x times x times pi plus x we get an r at approximation for sine x and again for reasons that will become apparent later we're going to divide by x on both sides again giving us sine x over x is approximately equal to pi minus x times pi plus x now we could get a much better approximation by multiplying this by some constant to give our approximation function the right amplitude it turns out this happens at one over pi squared and as we had more factors we must keep dividing by whatever root we added for example if we added 2 pi minus x times 2 pi plus x we must divide by 1 over 2 pi twice leaving our approximated function being 1 over 4 pi to the fourth times pi minus x times pi plus x times 2 pi minus x times 2 pi plus x more simply we can distribute these 1 over something pi's accordingly and we can get sine x over x is equal to 1 plus x over pi times 1 minus x over pi times 1 plus x over 2 pi times 1 minus x over 2 pi and so on now we can multiply some of these adjacent factors together and we get a bunch of differences of perfect squares doing that we'll get sine x over x is equal to 1 minus x squared over pi squared times 1 minus x squared over 4 pi squared times 1 minus x squared over 9 pi squared and so on and now that we've written sine x over x this way let's just focus on what happens to the coefficient of x squared when we distribute these differences of perfect squares doing that we'll get negative one over pi squared plus one over four pi squared plus one over nine pi squared and so on all times x squared now this is starting to seem a little bit familiar let's start by factoring out one over pi squared and see where that brings us all right so factoring out one over pi squared gives us negative one over pi squared times one over one plus one over four plus one over nine and so on times x squared all right now this is looking super familiar this looks like our problem from the beginning the sum of the reciprocal of the squares but how does this help us though all we have is the original problem we were trying to solve plus some other stuff well let's try comparing our original maclaurin series for sine x over x to the infinite product function for sine x over x well since these two functions are equal we can compare the coefficients of the different terms of x comparing the coefficient of x squared in our infinite product formula to the coefficient of x squared in the maclaurin series we get negative one over pi squared times one over one plus one over four plus one over nine and so on times x squared is equal to negative one over six x squared now all we have to do from here is divide both sides by x squared multiply the pi squared over to the other side divide by negative one and we get one over one squared plus one over two squared plus one over three squared and so on is equal to pi squared over six and that is how euler found that zeta of two was exactly equal to pi squared divided by 6. now this isn't the only thing that euler's managed to pull from the zeta function in addition to solving the basil problem he's also found a way to rewrite the zeta function as an infinite product he wrote the data function as an infinite product of 1 over one minus the reciprocal of the prime numbers raised to some complex exponent s a much cleaner way to write an infinite product is by using a capital pi normally we see sigma which represents repeated summation whereas pi represents repeated multiplication euler's product formula is not only useful since it gives us another way to write the zeta function but it also connects the zeta function to the prime numbers an interesting fact about this product formula is that we can use it to determine the probability that two randomly chosen integers are relatively prime let's let a and b each equal a randomly selected positive integer and we want to find the probability that they are relatively prime and this just means that a and b don't have common factors for instance 15 and 8 would be relatively prime since 15 doesn't share any of the same factors as eight let's just start with asking what is the probability that our numbers aren't divisible by two well since one half of all numbers are even the probability that a or b isn't divisible by two would be one half but what is the probability that both a and b aren't divisible by two well since the probability that both a and b are divisible by two is one half times one-half the probability that both aren't divisible by one-half would be one minus one over two squared we can see this follows the same pattern with being divisible by three five and so on so we know that the probability that a and b aren't divisible by two is one minus one over two squared but what would the probability be that a and b are not divisible by two or three well we would just multiply this by the probability that both a and b are not divisible by three carrying this on with divisibility by 5 7 11 and so on when carrying this on forever we can rewrite this as an infinite product of one minus one over p squared where p is the prime numbers going from 2 to infinity now this definitely should look a bit familiar this is actually the reciprocal of euler's product formula for zeta function evaluated at s equals 2. and we know from earlier that zeta of 2 geniusly solved by euler is equal to pi squared divided by six the reciprocal of this would be six over pi squared and thus we find that the probability of two randomly selected positive integers being relatively prime is exactly equal to six over pi squared now before delving too much deeper into the zeta function it's probably a good idea to understand what it even means to raise a number to a complex exponent let's start by raising one half to the power of i where i is the square root of negative one to get a value for this we need to start by rewriting one half to the i power as e to the power of the natural log of one half to the i power which is just equal to e to the power of i times the natural log of a half now using euler's function e to the ix equals cosine x plus i sine x we find that e to the i ln of a half is equal to cosine ln of a half plus i sine ln of a half the value we get for this is around 0.769 minus 0.639 i now this is going to be a bit confusing but the magnitude or distance from the origin of cosine x plus i sine x in the complex plane is equal to one this means that every point on e to the x i which is just equal to cosine x plus i sine x lies on the unit circle in the complex plane since the unit circle is just the set of all points whose distance from the origin is one for a value such as one half to the two plus i we can rewrite this as one half squared times one half to the i power the real part one half squared is rotated to be in line with one half to the i power the point that lies on the unit circle the value of one half squared is not changed its magnitude stays the same the point is only rotated and quick side note if we plug in pi to euler's formula we actually get e to the i pi is equal to negative one which is pretty cool the raymond hypothesis concerns the raymond's zeta function and its zeros it is known that all non-trivial zeros trivial zeros being roots with the real part is negative even integer of the function lie where the real part of s is between zero and one it hypothesizes that all of the roots of the riemann zeta function occur where the real part is one half a lot of books in the work of mathematicians are written on the assumption that the raymond hypothesis is true and if the hypothesis can be proven the clay mathematics institute will give the person who solved it a million dollars as a part of their millennium prize problems if it turns out to be correct mathematicians will have a better idea of how prime numbers are distributed which is a large part of number theory bernard raymond was born in 1826 he grew up in a