Home #1 Mathematics What are the 'Music of the Primes'? (Part I) ... by Giovanni A. Orlando.
What are the 'Music of the Primes'? (Part I) ... by Giovanni A. Orlando. PDF Print E-mail
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Monday, 13 February 2012 00:00

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Greetings ... in the day of the Will of God ... the Royal Blue day ...

    And me ... Giovanni ... your fellow friend ... agree with Prof. Edmund Landau, a German from Hebrew origins expert in Number Theory ... contemporaneous of Englishman ... Prof. Godfrey Harold Hardy ... one of the most prominent expert in Number Theory in past century ... I agree with Prof. Landau ... to exclude ... the number 1 from the primes ... because 1 is not prime.

    ... but I include in the previous figure ... because you have '1' like a prime ... and is not.

   To conclude the introduction of Professor Landau, I need to comment that ... Professor Landau publish some excellent Books in German ... about Number Theory, as well about the Fundamentals (Grundlagen der Analysis)

    Between the books Professor Landau publish in Number Theory we have two classical:

   I have these books from the age of sixteen (and the Hardy book, An Introduction still from before) ... but these books never has been translated into English ... only Vol 1+Part I from the Vorlesungen and I consider that these volumes in English will be a great contribute.

    To-day I want to gift you the First Part of 'The Music of Primes' ...

    This time I am using the book ... The Music of Primes ... by Marcus Du Sautoy.

Graphically ... you can watch this movie ...

 

 

In fact, extending the Zeros of the Riemann Zeta Function.

 

 

We have that all these Zeros ... looks to line on the same plane ...

 

 

The Plane z = 1/2 is that plane ...

 

For a better explanation you can read ...

 

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CHAPTER   FOUR

The Riemann Hypothesis: From Random Primes to Orderly

Zeros The Riemann Hypothesis is a mathematical statement that you can decompose the primes into music. That the primes have music in them is a poetic way of describing this mathematical theorem. However, it's highly  post-modern music.

Michael Berry. University of Bristol

 

Riemann had found a passageway from the familiar world of numbers into a mathematics which would have seemed utterly alien to the Greeks who had studied prime numbers two thousand years before. He had inno­cently mixed imaginary numbers with his zeta function and discovered, like some mathematical alchemist, the mathematical treasure emerging from this admixture of elements that generations had been searching for. He had crammed his ideas into a ten-page paper, but was fully aware that his ideas would open up radically new vistas on the primes.

 

Riemann's ability to unleash the full power of the zeta function stems from critical discoveries he made during his Berlin years and in his later doctoral studies in Göttingen. What had so impressed Gauss while he was examining Riemann's thesis was the strong geometric intuition that the young mathematician showed when he was feeding functions with imagin­ary numbers. After all, Gauss had capitalised on his own private mental picture to map out these imaginary numbers before he dismantled the conceptual scaffolding. The starting point for Riemann's theory of these imaginary functions had been Cauchy's work, and for Cauchy a function was defined by an equation. Riemann had now added the idea that even if the equation was the starting point, it was the geometry of the graph defined by the equation that really mattered.

 

The problem is that the complete graph of a function fed with imagin­ary numbers is not something that is possible to draw. To illustrate his graph, Riemann needed to work in four dimensions. What do mathema­ticians mean by a fourth dimension? Those who have read cosmologists such as Stephen Hawking might well reply 'time'. The truth is that we use dimensions to keep track of anything we might be interested in. In physics there are three dimensions for space and a fourth dimension for time. Economists who wish to investigate the relationship between interest rates, inflation, unemployment and the national debt can interpret the economy as a landscape in four dimensions. As they trek uphill in the direction interest rates, they will be exploring what happens to the econ­omy in the other directions. Although we can't actually draw a picture of this four-dimensional model of the economy, it is still a landscape whose hills and troughs we can analyze.

 

For Riemann, the zeta function was similarly described by a landscape that existed in four dimensions. There were two dimensions to keep track of the coordinates of the imaginary numbers being fed into the zeta func­tion. The third and fourth dimensions could then be used to record the two coordinates describing the imaginary number output by the function.

 

The trouble is that we humans exist in three spatial dimensions and so cannot rely on our visual world for a perception of this new 'imaginary graph'. Mathematicians have used the language of mathematics to train their mind's eye to help them 'see' such structures. But if you lack such mathematical lenses, there are still ways to help you to grasp these higher-dimensional worlds. Looking at shadows is one of the best ways to understand them. Our shadow is a two-dimensional picture of our three-dimensional body. From some perspectives the shadow provides little information, but from side-on, for example, a silhouette can give us enough information about the person in three dimensions for us to recognize their face. In a similar way, we can construct a three-dimensional shadow of the four-dimensional landscape that Riemann built using the zeta function which retains enough information for us to understand Riemann's ideas.

Gauss's two-dimensional map of imaginary numbers charts the numbers that we shall feed into the zeta function. The north-south axis keeps track of how many steps we take in the imaginary direction, whilst the east-west axis charts the real numbers. We can lay this map out flat on a table. What we want to do is to create a physical landscape situated in the space above this map. The shadow of the zeta function will then turn into a physical object whose peaks and valleys we can explore.

The height above each imaginary number on the map should record the result of feeding that number into the zeta function. Some information is inevitably lost in the plotting of such a landscape, just as a shadow shows very limited detail of a three-dimensional object. By turning this object we get different shadows which reveal different aspects of the object.

Similarly, we have a number of choices for what to record as the height of the landscape above each imaginary number in the map on the table top. There is, however, one choice of shadow which retains enough information to allow us to understand Riemann's revelation.

 

The zeta landscape - Riemann discovered how to continue this picture into a new land to the west.

 

It is a perspective that helped Riemann in his journey through this looking-glass world. So what does this particular three-dimensional shadow of the zeta function look like?

 

As Riemann began to explore this landscape, he came across several key features. Standing in the landscape and looking towards the east, the zeta landscape levelled out to a smooth plane 1 unit high above sea level. If Riemann turned round and started walking west, he saw a ridge of un­dulating hills running from north to south. The peaks of these hills were all located above the line that crossed the east-west axis through the number 1.

 

Above this intersection at the number 1 there was a towering peak which climbed into the heavens. It was, in fact, infinitely high. As Euler had learned, feeding the number 1 into the zeta function gives an output which spirals off to infinity. Heading north or south from this infinite peak, Riemann encountered other peaks. None of these peaks, however, were infinitely high. The first peak occurred at just under 10 steps north at the imaginary number 1 + (9.986. . .) i and was only about 1.4 units high.

 End First Part.

Thanks,

Giovanni A. Orlando.

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Last Updated on Monday, 13 February 2012 18:06