# Curve of zeta(0.5 + i t) : "Dense" on complex plane?

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• Swamp Thing
In summary, there is a discussion on MathOverflow regarding a conjecture about the "density" of the curve of the zeta function on the complex plane. The term "dense" is defined as having at least one point inside any arbitrarily small circle drawn on the plane. The argument presented suggests that the set of zetas is not dense, but a counterexample is provided. The flaw in the argument is that it does not consider the possibility of curves that can fill the entire plane, known as "space-filling curves." To prove that the curve is not dense, it would need to be shown that there is at least one area that is never further subdivided, regardless of the value of t.
Swamp Thing
This is a discussion on MathOverflow where a conjecture is discussed that the curve of ##\zeta(0.5+it)## is "dense" on the complex plane.
https://mathoverflow.net/questions/...of-riemann-zeta-function-on-the-critical-line

From a couple of sources, e.g. www.reddit.com/r/math/comments/8c9uh7/dense_sets_in_the_complex_plane/
I learned that "dense" means this: No matter how small a circle you draw and no matter where it's centered, the dense set has at least one point inside the circle.

From the following argument, it appears that the set of zetas is not dense. What is the flaw in it (if any)?

Let's look at the curve from t=0 up to, say, the 100th zero. We see that it divides the plane into a lot of areas that look like diamonds or like stretched-out diamonds, or crescent like slivers. Let's pick a point in the middle of one of these areas and draw a circle that is entirely within it. Now let's extend our curve from the 100th zero to the 101st zero. If the extended curve doesn't pass through our circle, we're still good. But if the curve now cuts our circle, it means that our chosen diamond subdivision has been further subdivided. So we just dodge away from the curve by choosing a new point within one of the new smaller subdivisions. We choose a circle that lies within that, rinse and repeat.

Thus if we extend the curve through any number of zeros, we can keep dodging into the smaller and smaller subdivisions (when required). If we take this to infinity, we get a point that is the sum of our original position, plus a potentially infinite convergent series of evasive moves. This complex number is, in one sense, not on the curve for t -> infinity.

I'm not sure if the above is correct. Maybe I've fallen into a sort of Zeno's paradox, and have wrongly interpreted the definition of dense. Maybe the answer depends on the relative rate at which t has to grow, versus the rate at which our circle shrinks?

Any clarifications greatly appreciated.

You seem to be claiming that a continuous curve can never be dense in the plane (since you don't use any fact about the zeta function other than it being continuous). Here is a counterexample: Let ##q_1,q_2,\ldots## be an enumeration of rational points in the plane (i.e. both coordinates are rational). Then take a piecewise-linear curve that starts at ##q_1##, goes to ##q_2##, etc. Then every rational point lies on the curve, so the curve is dense in the plane.

Edit: You can in fact have curves whose image is the entire plane (not just dense). Search "space-filling curves" for explicit constructions.

Last edited:
FactChecker and fresh_42
I think I get it now -- to prove that the curve is not dense, we would have to prove that there is at least one area that never gets further subdivided, however high we take t?

## 1. What is the "Curve of zeta(0.5 + i t) : "Dense" on complex plane?

The "Curve of zeta(0.5 + i t) : "Dense" on complex plane is a plot of the values of the Riemann zeta function at the point 0.5 + i t, where t is a real number. This curve is often referred to as "dense" because it shows a high concentration of points on the complex plane.

## 2. What is the significance of the "dense" curve on the complex plane?

The "dense" curve on the complex plane is significant because it is related to the distribution of prime numbers. The Riemann zeta function is closely connected to the prime counting function, and the "dense" curve can provide insights into the behavior of prime numbers.

## 3. How is the "dense" curve calculated?

The "dense" curve is calculated by evaluating the Riemann zeta function at various points on the complex plane. This can be done using numerical methods, such as the Euler-Maclaurin formula, or by using computer algorithms.

## 4. What is the Riemann Hypothesis and how does it relate to the "dense" curve?

The Riemann Hypothesis is a conjecture in mathematics that states all non-trivial zeros of the Riemann zeta function lie on the critical line 0.5 + i t. If this hypothesis is true, then the "dense" curve would have a very specific shape, known as the "critical line." However, the Riemann Hypothesis has yet to be proven, and the shape of the "dense" curve remains a mystery.

## 5. Why is the "dense" curve important in mathematics?

The "dense" curve is important in mathematics because it is connected to many unsolved problems, such as the Riemann Hypothesis and the distribution of prime numbers. It also has applications in other areas of mathematics, such as number theory and complex analysis. Studying the "dense" curve can provide valuable insights and potentially lead to new discoveries in these fields.

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