#naturalnumbers

re Riemann's zeta function

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  • Prompt: Imagine you have a list of natural numbers (1, 2, 3, 4, ...). Riemann's zeta function takes a number 's' and sums the reciprocals of all the numbers in that list raised to the power of 's'. For example, zeta(2) = 1/1^2 + 1/2^2 + 1/3^2 + ... (sum of reciprocals of squares of natural numbers). This function is interesting because it helps solve many problems related to prime numbers, which are the building blocks of all natural numbers. However, the zeta function is not defined for all values of 's'. It gets a bit tricky at 's' = 1. The trickiness at s = 1 in the Riemann zeta function can be visualized by imagining a teeter-totter: Think about the sum for zeta(2) again: 1/1^2 + 1/2^2 + 1/3^2 + .... Each term gets smaller and smaller as you go further down the list. This makes the sum well-behaved and eventually converges to a specific value. But now, imagine s = 1. The sum becomes 1/1 + 1/2 + 1/3 + .... Here, each term is either 1 or bigger than 1. It's like putting only heavy weights on one side of the teeter-totter. The sum keeps getting bigger and bigger, never settling on a specific value. Mathematicians call this situation "divergence." There are ways to define the zeta function for other values of 's' even though the simple sum diverges at 1. It's like using advanced techniques to balance the teeter-totter so it makes sense even with those heavy weights. But understanding that initial imbalance at s = 1 is the first hurdle.
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