#riemann zeta function

[a-number-a-day]

Today's number is Apéry's constant

Most famous constants announce themselves immediately. For example, π appears wherever circles show up, e emerges from growth and calculus, and so on. But there's a stranger hiding deep within this infinite series:

ζ(3) = 1 + 1 / 2^3 + 1 / 3^3 + 1 / 4^3 + . . .

The number this sum converges to is called Apéry's constant, and despite looking innocent, it resisted proof for centuries. Mathematicians strongly suspected that it was irrational but nobody could prove it until 1978.

The Zeta Function Apéry's constant comes from one of the most important objects in mathematics: the Riemann zeta function.

For real numbers s > 1,

ζ(s) = ∑_{n=1}^∞ 1 / n^s

(the infinite series of 1 / 1 ^ s + 1 / 2 ^ s + 1 / 3 ^ s + . . . )

At first glance, this is just an infinite sum. But the zeta function secretly connects prime numbers, complex analysis, quantum physics, probability, cryptography, and the distribution of the primes themselves.

Some values are beautifully understood. For example,

ζ(2) = π^2 / 6

ζ(4) = π^4 / 90

In fact, every even positive integer produces a formula involving powers of π. But the odd inputs are another story.

Nobody knows a comparably elegant formula for

ζ(3), ζ(5), ζ(7), . . .

These numbers are mysterious, and ζ(3) became the first "battleground".

Roger Apéry's Bombshell

Numerically, Apéry's constant equals approximately

1.202056903159694...

The question sounds deceptively simple: Is this number rational? For over 200 years, nobody knew. That's remarkable because Euler had solved the analogous problem for ζ(2) in the 1700s.

In 1978, French Mathematician Roger Apéry announced that ζ(3) was irrational in a lecture. The announcement was met with criticism in part because Apéry was relatively unknown at the time. He also gave the lecture in French, made jokes throughout, and omitted several key explanations needed to follow the proof..

For example, there was an equation at the beginning of his lecture that no one knew but formed the core of his proof. When asked where this equation came from, Apéry is said to have answered "They grow in my garden," which was said to have caused many in the audience to stand up and leave the room.

However, someone in attendance had an electronic calculator (uncommon at the time) and with a short program, checked Apéry's equation and found it correct.

The equation in question is below, which was an unknown series representation of ζ(3) at the time:

With this expression, he was able to use a condition for irrationality that German mathematician Gustav Lejeune Dirichlet had derived in the 19th century. It states that a number χ is irrational if there are an infinite number of integers p and q, so that the following inequality is satisfied:

Here, c and δ denote constant values. Although the formula looks complicated, it basically means that χ can be approximated by fractions, but there is no fractional number that corresponds to χ exactly. Apéry succeeded in deriving this inequality for ζ(3), and thus the number is irrational.

In simpler terms, Apéry's proof constructed two sequences of integers:

a_n and b_n

such that

a_n / b_n

approximate ζ(3) far too well for a rational number.

This is the key philosophical idea. If a number is rational, there are limits to how accurately fractions can approximate it without eventually becoming exact. Apéry built approximations that violated those limits.

The machinery involved strange recursive sequences and combinatorial identities that seemed to come out of nowhere.

Even now, many mathematicians describe the proof as "magical".

To honor his work, the value of ζ(3) now bears his name and is known as Apéry's constant. This doesn't answer all of the questions associated with the number, however. We are still looking for a clear numerical value for ζ(3) that can be expressed with known constants, much in the same way ζ(2) is.

But regardless, Apéry's constant feels like an accident. It emerges from a simple series, has no known simple closed form, and required centuries to understand even partially. And yet it keeps appearing across math and physics like a recurring character in a story nobody fully understands.

Not every important mathematical object arrives polished and symmetrical. But that's part of what makes ζ(3) beautiful.

[prime-numbers-official]

guys stop asking us about the zeta zeroes. we promise you that we have no affiliation with that infinite polynomial. stop looking. who cares about the zeta zeroes. this is not a coverup.

[brightgreendandelions]

hey you're super cool! now spread this to a number of blogs equal to the amount of non-trivial zeros of the Riemann zeta function with a real part not equal to 1/2 (with proof)

no anon you're doing it wrong! >:[

if you want to design a mind virus to solve the Riemann hypothesis you have to make the virus spread, and after it spread, the people affected must try solving the problem

[primepatterns]

Riemann zeta function: leveraging main bulb repetition to speed-up fractal rendering

In my previous post, I demonstrated how any fixed image of a certain size centred on the origin can be deformed and repeated in arbitrary regions of the complex plane by the action of the Riemann zeta function.

In practice, for each point, s, in the relevant region of the complex plane, ζ(s) is calculated. If the real and imaginary parts of ζ(s) are within the bounds of the area covered by the fixed image, the colour of the pixel at the {x,y} value {Re(ζ(s), Im( ζ(s))} can be used to colour the pixel at s.

If the fixed image is a high resolution image of the iteration fractal of the Riemann zeta function, we can avoid iterating every point, s, in the target region of the complex plane by using the fixed image as a "lookup table".

Here is the resultant plot for the region of the complex plane covering -2 Re(s) < 5 and 80 < Im(s) < 100 at a resolution of 100 pixels per unit. The image has been rotated clockwise by 90° such that the imaginary axis runs from left to right. The image is derived from a 5001 x 5001 pixel fixed image "lookup table" whose points, f, are in the following ranges: -25 < Re(f) < 25 and -25 Im(f) < 25, also at a resolution of 100 pixels per unit:

-2 < Re(s) < 5, -80 < Im(s) < 100, 100 pixels per unit

It is clear that, for values of s where Re(s) > 0, there is insufficient resolution in the lookup table to give decent resolution in the final image. In the positive real half plane, the value of ζ(s) tends to 1. On that basis, it makes sense to increase the resolution of the lookup table close to the pole of the Riemann zeta function at f = 1.

However, given that the resolution in the rest of the generated image is reasonable, there is no need to create a very large lookup table with a resolution of, say, 1000 pixels per unit. Instead, we can create a series of "nested" lookup tables of increasing resolution centred on the pole, and programmatically decide which to consult after the value of ζ(s) has been calculated.

Here is the resultant plot for same region of the complex plane as above, but derived from a series of four 5001 x 5001 pixel fixed image lookup tables whose points, f, are centred over the pole and which have resolutions of 100, 1000, 10000 and 100000 pixels per unit:

-2 < Re(s) < 5, -80 < Im(s) < 100, 100 pixels per unit

The resolution of the region where Re(s) > 0 is much improved.

Here are the fixed image lookup tables used to generate the above image. Note that they are all centred on the pole at f = 1:

-24.0 < Re(f) < 26.0, -25.0 < Im(f) < 25.0, 100 pixels per unit

-1.50 < Re(f) < 3.50, -2.50 < Im(f) < 2.50, 1000 pixels per unit

0.75 < Re(f) < 1.25, -0.25 < Im(f) < 0.25, 10000 pixels per unit

0.975 < Re(f) < 1.025, -0.025 < Im(f) < 0.025, 100000 pixels per unit

These tables are good for images where Re(s) < 16. Experiments suggest that a further fixed image lookup table with a resolution of 1000000 can extend the scope even further into the positive real half plane. However, I am typically interested in fractal images confined to the region where -2 < Re(s) < 5, so I tend to use only the above four fixed image lookup tables.

Another obvious limitation of using fixed image lookup tables to avoid formal iteration of each point, s, is that the values of ζ(s) where Re(s) < 0 are far too large to be accommodated by any reasonable lookup table. Pixels outside the bounds of the lowest resolution lookup table are therefore coloured black.

However, one aspect of these regions is that iteration is not very costly programmatically provided that appropriate bail-out limits are applied. The vast majority of points in this region blow-up on iteration, and they do so quickly.

Therefore, because only a few iterations are required for the majority of pixels outside the bounds of the lookup tables, it is a trivial matter to provide a function which enables the user to choose to formally iterate any pixel that falls outside the bounds of the lookup tables in order to "fill in" the black regions without too much impact on resolution time.

Here are some resultant images with and without fill in:

-2 < Re(s) < 5, 80 < Im(s) < 100, 100 pixels per unit, without fill-in

-2 < Re(s) < 5, 80 < Im(s) < 100, 100 pixels per unit, with fill-in

-2 < Re(s) < 5, 980 < Im(s) < 1000, 100 pixels per unit, without fill-in

-2 < Re(s) < 5, 980 < Im(s) < 1000, 100 pixels per unit, with fill-in

-2 < Re(s) < 5, 9980 < Im(s) < 10000, 100 pixels per unit, without fill-in

-2 < Re(s) < 5, 9980 < Im(s) < 10000, 100 pixels per unit, with fill-in

-2 < Re(s) < 5, 99980 < Im(s) < 100000, 100 pixels per unit, without fill-in

-2 < Re(s) < 5, 99980 < Im(s) < 100000, 100 pixels per unit, with fill-in

-2 < Re(s) < 5, 999980 < Im(s) < 1000000, 100 pixels per unit, without fill-in

-2 < Re(s) < 5, 999980 < Im(s) < 1000000, 100 pixels per unit, with fill-in

Using fixed image lookup tables leads to a reasonable speed up in rendering performance and good reproducibility. The resultant images are essentially indistinguishable from those generated by zeta_machine.

The rendering of images very high up the imaginary axis is still painfully slow, however, because for every point, s, ζ(s) must be calculated at least once, and the Euler Maclaurin summation requires a partial sum to be calculated with a number of terms in the order of |s|.

[fruitsoftheweb]

“It’s also possible to do surface plots. Using the Riemann zeta function again, a surface plot looks as follows“

[mymathematics]

Visualizing the Riemann zeta function and analytic continuation

Zeta function

Mathematics

[teratosapphic]

hey mathy side of tumblr

since  ζ(-1) is the infinite sum of all natural natural numbers, and also equals -1/12, would 2 * ζ(-1) = -1/6, -1/12, or -1/24?

[primepatterns]

Zeta functions: main bulb repetition in iteration fractals

In previous posts on this page, and in a recent talk on this subject, I have remarked that the iteration fractal in the neighbourhood of each non-trivial zero of the Riemann zeta function is a scaled and distorted reproduction of the main bulb.

One way of evidencing this phenomenon is to generate the fractal for each starting value, s, in a random section of the complex plane and, for each value of s where the real and imaginary parts of ζ(s) are both within some fixed distance from the origin, to place a pixel at that point with the same colour as the colour ascribed to s in the fractal image.

For example, the following iteration fractal image covers the section of the complex plane between -2 ≤ Re(s) ≤ 13 and 100 ≤ Im(s) ≤ 150, with a resolution of 100 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right:

The image contains ‭7,506,501‬ pixels. For many of the individual starting values, s, the absolute values of the real and imaginary parts of ζ(s) are less than 25. If each qualifying pixel is plotted in a simple x,y plot covering the section of the complex plane between -25 ≤ Re(s) ≤ 25 and -25 ≤ Im(s) ≤ 25, using the colour of the pixel ascribed to the starting value, s, in the above image, the result is as follows:

It is clear that the overall image corresponds to the main bulb of the iteration fractal, but the density of coverage away from the origin decreases rapidly.

The same logic can be applied to Hurwitz zeta functions, ζ(s, a), with rational a.

For example, the following iteration fractal image for the Hurwitz zeta function ζ(s, 0.75) covers the section of the complex plane between -2 ≤ Re(s) ≤ 13 and 100 ≤ Im(s) ≤ 150, with a resolution of 100 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right:

As before, for many of the individual starting values, s, the absolute values of the real and imaginary parts of ζ(s, 0.75) are less than 25. If each qualifying pixel is plotted in a simple x,y plot covering the section of the complex plane between -25 ≤ Re(s) ≤ 25 and -25 ≤ Im(s) ≤ 25, using the colour of the pixel ascribed to the starting value, s, in the above image, the result is as follows:

Again, it is clear that the overall image corresponds to the main bulb of the iteration fractal. The density of coverage is higher than for the Riemann zeta function.

It is also possible to work in the opposite direction.

For example, the following iteration fractal image for the Hurwitz zeta function ζ(s, 0.75) covers the section of the complex plane between -25 ≤ Re(s) ≤ 25 and 25 ≤ Im(s) ≤ 25, with a resolution of 40 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right:

The absolute value of the real and imaginary parts of ζ(s, 0.75) for each starting value, s, in a given section of the complex plane may be less 25. If so, we can colour the pixel corresponding to s using the corresponding pixel colour from the above image.

The result for the section of the complex plane covering -2 ≤ Re(s) ≤ 13 and 100 ≤ Im(s) ≤ 150, with a resolution of 100 pixels per unit, is as follows:

The image constitutes a reasonable approximation of the directly calculated fractal (see above) and takes considerably less time to generate. The black areas above and below the image correspond to places where the real and/or imaginary parts of ζ(s, 0.75) are greater than 25.

Another way to visualise this scaling and distortion phenomenon caused by zeta functions is to repeat the previous experiment with an arbitrary geometric design.

For example, the following image corresponds to the section of the complex plane covering -25 ≤ Re(s) ≤ 25 and 25 ≤ Im(s) ≤ 25, with a resolution of 40 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right:

The following image is formed by calculating ζ(s) for each staring value, s, in the section of the complex plane covering -2 ≤ Re(s) ≤ 13 and 200 ≤ Im(s) ≤ 220 and colouring those pixels where the absolute value of the real and imaginary parts of ζ(s) are less than 25 using the corresponding pixel colour from the above image. The result is as follows:

The overall pattern is preserved with repetition, but it is clear that the green and yellow sections are twisted such that they face upwards rather than their downward orientation in the original image.

Here is the corresponding image for ζ(s, 0.35):

It is clear that, not only are the green and yellow sections twisted above, there is a repetition below where the blue and red sections are twisted such that they face downwards rather than their upward orientation in the original image.

Here is the corresponding image for ζ(s, 0.75):

The repetition in the lower section has persisted. Additional experiments indicate that this repetition below is a feature of all Hurwitz zeta functions ζ(s, a) with rational a.

The same kind of distortion can be applied to any image arbitrarily centred on the origin in the complex plane. The results are particularly disconcerting when the original image is a human face. For example, here is a computer generated image of a human face copied from the website thispersondoesnotexist.com:

Here is the distortion caused by ζ(s, 0.35) in the section of the complex plane covering -2 ≤ Re(s) ≤ 13 and 200 ≤ Im(s) ≤ 220:

Here is the corresponding distortion caused by ζ(s, 0.75):

The inversion / twisting, and upper and lower repetitions that are a feature of Hurwitz zeta functions are easy to discern.