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## A mathematician’s mind-2

CHAPTER 2

SEE THE INVISIBLE

Section 2.1. Method of Decomposing N

As the name of the chapter say, here I will write about how to see things that are not visible. This is best explained by the following example. Recall Lemma 1.2 from Chapter 1 which states that

Lemma 1.2. If the sequence $a_n$ is uniformly distributed modulo 1 then

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

Now let us rewrite the above result as an asymptotic formula

$\displaystyle{\sum_{r=1}^{n} 1. f(a_r) \sim n \int_{0}^{1}f(x)dx. }$

This form looks pretty much normal and fails to reveal any hidden pattern. But if we break the above asymptotic form one step further as shown below;

$\displaystyle{\sum_{r=1}^{n} 1. f(a_r) \sim \sum_{r=1}^{n}1. \int_{0}^{1}f(x)dx }$

then it begins to look more promising and starts revealing inner hidden patterns. Such a pattern should should immediately ignite a mathematical mind to ask oneself to ask how would the formula look if I replace the coefficient of $f(a_r)$; which is currently 1; by a general term say $d_r$. Once we reach this question, we know that we are on the right track; we have found possible clues to a hidden pattern that could lead to a missing mathematical formula.

At this point we have no idea how the actual formula; if at all it exists; would look like. So the best approach at this point would be to a reasonable guess based on intuition and logic. In the above example, it is not hard to see that a reasonable guess should be

$\displaystyle{\sum_{r=1}^{n} d_r f(a_r) \sim \sum_{r=1}^{n}d_r \int_{0}^{1}f(x)dx. }$

If it happens that more than one guess seems reasonable, take all of them for consideration for the time begin. Later we can eliminate the incorrect models in the verification step. A good mathematician should be a great guesser. If you can make ten good guess, probably one is right and you have found something to work on. Ramanujan himself relied very strongly on his intuition and made discoveries that probably would never have been made had he not possessed this ability.

Section 2.2. More on the method of decomposing N

Let us see one more example of the method of decomposing N this time slightly more advanced then in our previous example. The following result is well known.

Lemma 2.1If $f$ is monotonic and continuous and defined in $[1; n]$ then

$\displaystyle{\sum_{r=1}^{n}f(n) = \int_{1}^{n}f(x)dx + O(|f(n)| + |f(1)|.}$

Take a good look at this result and try to find out if there is any possibility of improvisation. Observe that $f(n)$ and the integral on the RHS both contain an $n$. Let us decompose this as the sum of n 1’s. We write

$\displaystyle{\sum_{r=1}^{n} 1.f(1+...+1) = \int_{1}^{1+...+1}f(x)dx + O(|f(1+...+1)| + |f(1)|).}$

Now lets us work with our intuition and guess. Let $d_r$ be a sequence of positive reals and let $\sum_{k=1}^{r} d_k = S_r$. The above decomposition of $n$ should at one point or another lead us to guess that there could be a result which would look like

$\displaystyle{\sum_{r=1}^{n} d_r f(S_r) = \int_{S_1}^{S_n}f(x)dx + O(|f(S_n)| + |f(S_1)|).}$

Later we shall see that we have guessed the integral on the RHS is correctly while the error term is slightly incorrect. Nonetheless we have made a good guess based on which we can begin working.

Section 2.3. Verify your formula numerically

Now that our observation and intuition have led us to guess some new mathematical results, we should verify or test our guesses with a few numerical examples. The reason for doing this is that in case of explicit formulas like in our examples, it is advisable to gain some confidence in our guesses before we attempt to prove them rigorously. Rigorous proof are usually harder and more time consuming and we want attempt a proof only when we have sufficient confidence in our guesses. Remember that while verification for any number of cases cases does not prove a formula, one counter example is sufficient to disprove the formula.

I verified my guesses using simple $d_r = 1/r, r^2$ and $f(a_r) = 1/(1+a_r ^2)$ where $a_r$ is the sequence of the fractional part of $r\sqrt 2$ which is uniformly distributed modulo one as shown my the Equidistribution Theorem. In the age of computers we can verify a formula easily by writing a program (PARI, Mathematica or even using MS Excel).

Section 2.4. Prove your formula rigorously

After successful verification of the formula, we can begin to construct a rigorous mathematical proof. Yes there is absolutely no guarantee that we will indeed find a proof. But there is bright side of this. The harder it is to prove something, the more likely it is that we have stumbled upon something important. If we are unable to find a rigorous proof but our formula passes all verification tests thenwe can conjecture the result and hope that someone else finds a rigorous proof of disproof of our result. This is how mathematicians usually come up with conjectures. Perhaps the most example of this method is the Prime Number Theorem. Based on numerical observation, young Gauss conjectured that the number of primes not exceeding x is approximately equal to $Li(x)$. Almost a century later, this conjecture was proved true by Hadamard and Charles de la Vallee Poussin.

Continuing with our example, after verifying the formulas, I became more confident that the formulas are indeed true and I began looking for a mathematical proof. This stage is the most important, not only because we might end up proving a new result but also because we may encounter many unforeseen cases that needs to be taken care of, especially the boundary conditions, error terms, exceptions etc. For example when I was working on the proof of the first formula (or first guess), I found that the result is not true if $d_r = \sin (r), 1/r^2$ etc. It was then then I realized that the series $\sum_{r=1}^{\infty}d_r$ should be divergent for the first formula. Similarly in the second formula, I had guessed the error term incorrectly and it was only during the proof stage that I could find the correct error term. Such intricate details might escape the verification stage and unless we are careful in the proof stage, our formula would be incomplete. Finally I could prove the following results rigorously; the details of which can be found in the paper ‘On a unified theory of numbers‘.

Lemma 2.2. If $d_r$ is a divergent series of positive terms and $a_r$ is uniformly distributed modulo 1 then

$\displaystyle{\sum_{r=1}^{n} d_r f(a_r) \sim \sum_{r=1}^{n}d_r \int_{0}^{1}f(x)dx. }$

Notice that with Lemma 2.1, we have generalized Lemma 1.2.

Lemma 2.3If $f$ is monotonic and continuous and defined in $[1; n]$ and $d_r$ is a series of positive terms such that $\sum_{k=1}^{r} d_k = S_r$; then

$\displaystyle{\sum_{r=1}^{n} d_r f(S_r) = \int_{S_1}^{S_n}f(x)dx + O(\delta (n))}$

where

$\displaystyle{\delta (n) = d_n |f(S_n)| + d_1 |f(S_1)| + \int _{1}^{n} \frac{d(d_y)}{dy} S(y)dy.}$

Example.

$\displaystyle{\lim_{n \rightarrow \infty} \sum_{r < n} \frac{d_r}{S_n\sqrt{\ln S_n - \ln S_r}} = \sqrt{\pi}}.$

$\displaystyle{\sum_{r \le n}{p_r}^{a}({p_1}^{a}+{p_2}^{a} + \ldots + {p_r}^{a})^{b}\sim \frac{n^{b+1} p_n^{ab+a}}{(b+1)(a+1)^{b+1}}}.$

Notes at the end of Chapter 2

In this chapter we saw how breaking numbers using the method of decomposition throws light on how well know results in mathematics could be improved. Writing n as $\sum_{r=1}^{n} 1$ is an explicit example of this. This does not mean that this is the only trick. What the reader should grasp here is the idea of decomposition. A mathematical explorer should always try to visualize results by improvising and by breaking the results into simpler forms in one or more ways.

## A mathematician’s mind-1

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?

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.

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.