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A mathematician’s mind-1

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I have always been thinking how mathematicians make discoveries. They come up with beautiful theorem and formulas that were unknown to mankind an hour ago until they discovered it first. Unlike the poets and writers, mathematicians cannot cook some stories or imagine some beautiful verses in the mind. So how does it work? When you are solving a problem you know where to begin, you know where to end and you will mostly have the tools that will help you solve the problem. Now let assume that you teacher enters to your class and instead of giving you a problem to solve from your course, asks you to discover something that you previously did not know. Now you don’t know where to begin where to end where to look for something new. So how do you go about making a mathematical discovery?

I am writing a series of article in this blog where I shall describe the thought process that I underwent as I made some interesting discoveries in analytical number theory. The purpose of this article is not to make a better problem solver but rather to encourage one to think like a mathematician who is on a journey of mathematical exploration and discovery. Therefore I shall not go into the precise technical details of the proofs of the results in this article but rather I shall give the sketch of the main ideas that led to these results and describe how one thought led to another and eventually culminated in a small theory comprising of many little discoveries along the way. Anyone interested in deriving the actual proof can easily reconstruct the proof from the central idea and thought process as well the descriptive details given for each result. The purpose of this article is to encourage the mathematical thought process that lead to new discoveries. In case there are any terminologies which I feel might be unfamiliar to my readers, I have provided hyperlink to a Wikipedia page on the same.

CHAPTER 1

BEGINNING WITH A SIMPLE IDEA

SECTION 1. OBSERVATION

Many discoveries begin with an initial observation. For example thousands of years ago somebody might have observed that sum of the square of the sides of a right triangle equals the square of the hypotenuse. This observation stood the test of repeated experiments and only after that it would have inspired some ancient genius to establish this observation as a theorem by finding a rigorous proof. Similarly someone would have observed that the ratio of the circumference of a circle to its diameter remains constant and this eventually led to the discovery of \pi. A keen observation power is therefore one of the prerequisites of making a mathematical discovery.

Many years ago when I first learned definite integrals as a school student, I developed an affinity for one particular theorem which allowed us to evaluate the limits of a sum using definite integrals. The formula was

Lemma 1. If f(x) is Riemann integrable in [0,1] then

\displaystyle{\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{r=1}^{n}f\Big (\frac{r}{n}\Big ) = \int_{0}^{1}f(x)dx}.

Three years ago I came across a type of sequence called equidistributed sequences or uniformly distributed sequences. There I found the following result theorem that had striking resemblance to Lemma 1.

Lemma 2. If the sequence a_n is uniformly distributed modulo 1 then

\displaystyle{\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{r=1}^{n}f(a_r) = \int_{0}^{1}f(x)dx}.

As soon as I saw this result on uniformly distributed sequences modulo 1, I knew that I had seen something like it before; the result in Lemma 1. Replacing r/n in Lemma 1 by a_n we get Lemma 2. The similarity between Lemma 1 and Lemma 2 should automatically make us wonder why they are so similar. So at this point what are the options that we have? Either it is a coincidence or there is some hidden mathematical relation that both these results are following.  Since we are like explorers searching to make a discovery we would want to believe that this similarity is not a coincidence but because of some hidden mathematical relation that we seek to unearth.

SECTION 2. EXPLAINING THE OBSERVATIONS

Once we have found an observation to work on the next step is to find a mathematical explanation for the observation. If we can find an explanation within the existing mathematics then our observation is a new theorem or an extension or a generalization of one of the areas of existing mathematics. However if no known mathematics can give a satisfactory explanation of our observation then we will have to invent new mathematics all together.

Continuing with our example, what are the possible thoughts that can strike us when we try to explain the similarity between Lemma 1 and Lemma 2? The possible hypothesis are:

  • Lemma 1 and Lemma 2 are two different representations of the same phenomenon.
  • The sequence 1/n, 2/n, … , n/n approaches uniform distribution modulo 1 as n \rightarrow \infty.

To test our hypothesis we go back to the definition of uniform distribution modulo 1 and test if this definition applies to the sequence 1/n, 2/n, … , n/n as n \rightarrow \infty and we find that this actually is the case and our hypothesis is true.

SECTION 3. BUILD THE THEORY

So we have discovered that the sequence 1/n, 2/n, … , n/n approaches uniform distribution modulo 1 as n \rightarrow \infty. We shall use this starting point and a guiding star in our exploration. We now know that the theme of our exploration is going to be centered around sequences uniformly distributed modulo 1. The next logical questions that strikes (or rather should strike) our mind are:

  • Yes the sequence r/n uniform distribution modulo 1 but what about the sequence r^2/n^2 or in general r^a/n^a?
  • This will lead us to formulate the question in the most general form, when is the sequence b_r/b_n uniformly distributed modulo 1?

Again applying the definition of uniform distribution modulo 1, we can easily prove the following result. Let [x] denote the greatest integer function. We have

Lemma 3. As n \rightarrow \infty the sequence b_r/b_n,  r=1, 2, ..., n approaches uniform distribution modulo 1 if

\displaystyle{\lim_{n \rightarrow \infty}\frac{b_{[nt]}}{b_n} = t}.

for every t, 0<t<1.

SECTION 4. APPLY YOUR THEORY

So far so good but now we need show that our theory is valuable by showing its applications. We want to find the sequence that satisfy Lemma 3. It is easy to see that Lemma 3 implies b_n = o(n^{1+\epsilon}) for every \epsilon >0. Hence b_n grows much slower than any quadratic polynomial function of n.  At this point we sit an recall all well know sequences that grow at this  rate.

We can find many such sequences but one particular sequence that we cannot afford to miss out is the sequence of prime numbers. Recall that the prime number theorem implies that p_n \sim n\ln n. We find that the sequence of primes satisfy Lemma 3. Similarly we can show that the sequence of composite numbers denoted by c_n also satisfy Lemma 3.

We have found a lot of interesting insights so far. But can we be more adventurous? Notice that if two or more sequence satisfy Lemma 3 then their linear combination will also satisfy Lemma 3. So we can put together all our results so far in the form a beautiful and elegant theorem.

Theorem 1. Let p_n be the n-th prime, c_n be the n-th composite number and let \alpha, \beta and \gamma be any three constants such that \alpha p_n + \beta c_n + \gamma n \ne 0 then

\displaystyle{\lim_{n \rightarrow \infty}\frac{1}{n} \sum_{r \le n} f \Big(\frac{\alpha p_r + \beta c_r + \gamma r}{\alpha p_n + \beta c_n + \gamma n}\Big) = \int_{0}^{1}f(x)dx}.

Thus we see that starting with basic theorem on definite integrals and a fundamental theorem on uniform distribution modulo one, we by properly directing the thought process have ended up with a non-trivial theorem connecting prime numbers, composite numbers and natural numbers. Now we can let our imaginations run wild and we can use our theorem to derive many interesting results on prime or composites. For example

Example 1.

\displaystyle{\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{r \le n}\frac{p_n ^{2}}{p_n ^{2}+ p_r ^{2}}=\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{r \le n}\frac{c_n ^{2}}{c_n ^{2}+ c_r ^{2}}=\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{r \le n}\frac{n ^{2}}{n ^{2}+ r ^{2}} = \frac{\pi}{4}}.

Example 2.

\displaystyle{\lim_{x \rightarrow \infty} \frac{1}{\pi (x)} \sum_{p \le x}\Big(\frac{p}{x}\Big)^{a-1} \ln^{b-1} \Big(\frac{cx}{p}\Big)= \frac{c^a}{a^b}\Gamma (b, a\ln c)}.

To read the second part of this post, click A mathematician’s mind-2.

NOTES AT THE END OF CHAPTER 1

In this chapter we saw how observing similarities between two different results of mathematics can reveal that both these results are special cases of a larger family of relations provides an ideal platform for beginning a mathematical exploration. In the second post on this topic in Chapter 2, we shall continue our exploration forward from here develop our theory further by extending our results to divergent series. The topic in this series of posts are based on my paper ‘On a unified theory of numbers‘ which I have co-authored with Prof. Marek Wolf, Institute of Theoritical Physics, Warsaw, Poland. Currently we are revising and re-formating our paper. If you spot any mathematical or typographical error please do let us know. Also any suggestion or comments to improve this blog are welcome.

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4 Comments

  1. Arun P says:

    It will hit you from nowhere when you are doing something totally unrelated.I am a novice and lack expertise to comment on it.But I had some glimpses occasionally,which are very trivial if you compare it with mathematical works.I will give one example.Once when I was in school I was sitting at the back bench as usual when some classes were going on.I scribbled 3,4,5 3^2+4^=5^2 and observed that 5^2+12^=13^2.Pattern suddenly came to me (2n+1)^2+(((2n+1)^2-1)/2)=(((2n+1)^2+1)/2)^2 and now I am able to generate an infinite set of pythagorean triples (x,y,z) such that the members in the set are linearly independent as f(n) where z-y=1.It is trivial to show that it is an identity using algebra.But this relation hit me without any effort.It was very vague.And I was very much convinced that this is true and valid for any n without even trying to expand it.It was a very wonderful experience.After observing the pattern I got very much interested in it.Somehow I felt that I can find another set (x,y,z) such that z-y=2.12,35,37 for instance came to me and instantly I generated (2n)^2+(((2n)^2/2)-2)/2)=(((2n)^2/2)+2)/2).In this case I did try to find out few examples.But this general formula hit from nowhere without any effort while I was watching some cricket match in school(I don’t remember exactly) .There was some sort of wishful thinking and conviction also.I am very poor problem solver especially under timebound and pressure conditions.I am very careless and reckless also.If it comes to me it comes.It never comes if I try to force myself and I get very frustrated,Partially this can be attributed to the fact that I have some mental disabilities like ADHD and bipolar depression and had undergone treatment for it.Usually when finding a new relation it automatically happens and get the feeling that this ought to be right without any effort.

  2. To me sometimes it hits completely without warning but other times I know where to look for it. I have completed the article so it will give a fair idea about the message that I want to convey. I have used an example from one of my works to show how form one simple thought we can reach totally different and unexpected results.

  3. SRINIVASA RAGHAVA says:

    Dear Nilotpal Sinha, Thanks for ur nice article. I had an improved version of the above theorem. Actually that type of problem I have found in some American Mathematical Monthly problems section.

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