![PI010101's tweet photo. Here is one surprisingly elegant equivalent reformulation of the Riemann Hypothesis: Let B be the pointwise maximal function on [0,1] satisfying B(0) = B(1) = 0 and 2B((x+y)/2) ≤ B(x) + B(y) + |x - y| for all x and y in [0,1]. Extend it periodically to all the real line.
As Grok correctly notes, this is Takagi function (or blancmange curve), with a fractal-like graph. It is continuous, nowhere differentiable, and less well-known than the Weierstrass function.
This function also emerged recently (and independently) in a joint work https://t.co/pgJw9MaEA1 with my student, Natanael Alpay, as a sharp lower bound for the L^1 norm of the square function applied to indicator functions of sets. I was unaware at the time that it is known as the Takagi function or of its connection to the Riemann Hypothesis. Large language models will undoubtedly accelerate scientific discoveries by forging connections across diverse fields of science.](https://pbs.twimg.com/media/G86cuY-WMAAM4o0.jpg)
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Here is one surprisingly elegant equivalent reformulation of the Riemann Hypothesis: Let B be the pointwise maximal function on [0,1] satisfying B(0) = B(1) = 0 and 2B((x+y)/2) ≤ B(x) + B(y) + |x - y| for all x and y in [0,1]. Extend it periodically to all the real line.
As Grok correctly notes, this is Takagi function (or blancmange curve), with a fractal-like graph. It is continuous, nowhere differentiable, and less well-known than the Weierstrass function.
This function also emerged recently (and independently) in a joint work https://t.co/pgJw9MaEA1 with my student, Natanael Alpay, as a sharp lower bound for the L^1 norm of the square function applied to indicator functions of sets. I was unaware at the time that it is known as the Takagi function or of its connection to the Riemann Hypothesis. Large language models will undoubtedly accelerate scientific discoveries by forging connections across diverse fields of science.
![PI010101's tweet photo. Here is one surprisingly elegant equivalent reformulation of the Riemann Hypothesis: Let B be the pointwise maximal function on [0,1] satisfying B(0) = B(1) = 0 and 2B((x+y)/2) ≤ B(x) + B(y) + |x - y| for all x and y in [0,1]. Extend it periodically to all the real line.
As Grok correctly notes, this is Takagi function (or blancmange curve), with a fractal-like graph. It is continuous, nowhere differentiable, and less well-known than the Weierstrass function.
This function also emerged recently (and independently) in a joint work https://t.co/pgJw9MaEA1 with my student, Natanael Alpay, as a sharp lower bound for the L^1 norm of the square function applied to indicator functions of sets. I was unaware at the time that it is known as the Takagi function or of its connection to the Riemann Hypothesis. Large language models will undoubtedly accelerate scientific discoveries by forging connections across diverse fields of science.](https://pbs.twimg.com/media/G86cwEeW0AAKGga.png)
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