What is your favorite Number? 1591521

While playing with the vowels of the English alphabet, I discovered the following facts about the mathematics of the vowels ---

The position occupied by the vowels in numbers are- a (1) e (5) i (9) o(15) and u (21). The numbers clearly show that all the vowel positions are odd numbers. It means there are no even vowels. The three vowel positions are divisible by 3. The two vowel positions are divisible by 5 and one vowel position is divisible by 7.

The other interesting play with the vowels is in terms of the scale. When all the vowel positions are written as a number, the number which is formed is 1591521. This is only divisible by 3 and none of the other 120549 available prime numbers less than 1591521.

$\frac{1591521}{3}=530507$, which is a prime number. It means the number formed is interesting to the number theorists.

When some manipulations are done on the digits of the number, this can give the scale, we are not sure up to which value but to a considerable large value. Here some manipulations mean we can arrange the digits of the number according to their order e.g. we can consider ($15$ or $91$ or $52$ ) or we can take ($59$ or $15$ or $21$) and so on but not the random ordering. we can construct a scale based on the digits available in $1591521$ as- $1=1$ (already there), $2=2$ (already there), $3=2+1$, $4=5-1$ or $2+1+1$, $5=5$ (already there),$ 6=5+1$;$7=5+2$; $8 =5+2+1$; $9= 5+5-1$; $10= 5+5$; $11=9+2$; $ 12=9+2+1 $ and so on.

By doing so, we can develop the scales continuously up to a large number using the available digits. Like some of the scales of large number are

 $43=59-15-1; 45=49-15+1; \\ 69=91-21-1; 70=91-21; \\  89=91-2; 100=91+5+2+1+1;\\120=21\times 5 +15;  200=21 \times 9 + 5+5+1 \\ $

and so on. This magical number seems interesting. Hence 1591521 is my best number for now.

Riemann Zeta Function

The Riemann zeta function, denoted by $\zeta(s)$ is a function of $s$ and given by
$\zeta(s)=\displaystyle \sum_{n=1}^\infty \frac{1}{n^s}=1+\frac{1}{2^s}+\frac{1}{3^s}+\cdots$,
 where $s$ is a complex variable and the function is absolutely convergent when the real value of s is greater than 1.

Riemann zeta function can also be expressed in terms of integral as
$\zeta(s)=\frac{1}{\Gamma(s)}\int_0^\infty \frac{x^{s-1}}{e^x-1}dx$

This is the convergent series for any $s > 1$.Consider the case when $s=2$, then
$\zeta(2)=\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}=1+\frac{1}{4}+\frac{1}{9}+\cdots = \frac{\pi^2}{6}$.

For $s=1$, it is divergent infinite series, called the harmonic series as given by
$\zeta(1)=\displaystyle \sum_{n=1}^\infty \frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots=\infty$.

For even positive integers, the Riemann zeta function can be expressed in terms of the Bernoulli number ($B_0=1, B_1=\pm\frac{1}{2}, B_2=\frac{1}{6}, B_3=0, B_4=\frac{-1}{30}, B_5=0, B_6=\frac{1}{42}, B_7=0, B_8=\frac{-1}{30}, B_9=0,\\ B_{10}=\frac{5}{66}, \cdots $ and so on) because all odd Bernoulli's number are zero except 1. More about Bernoulli's number are referred to  https://en.wikipedia.org/wiki/Bernoulli_number . The expression is given by
$\zeta(2n)= \frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}$ for any $n \ge 1$.

For negative integers, Riemann zeta function is also expressed in terms of Bernoulli number as
$\zeta(-n)=-\frac{B_{n+1}}{n+1}$ .
Hence $\zeta(-1)=1+2+3+4+\cdots= -\frac{B_2}{2}=-\frac{1}{12}$. This also explains the Grandi's series as we discussed earlier. In this case, the series is divergent and this specific infinite sum of natural numbers have been used to explain the string theory. More about this is available in the following link https://plus.maths.org/content/infinity-or-just-112.

Godel's Incompleteness Theorem

Godel's incompleteness theorem- a theorem that proves the "Un-Provable". A mathematical proof that it is possible or impossible to prove some given mathematical problems. Gödels theorem is the basis of modern mathematics to state that axioms exist without the formal proof. Kurt Gödel proposed two incompleteness theorems and these theorems are also one of the most important theorems in modern logic.

First incompleteness theorem states that for a consistence and formal system, there are statements which can be neither proved and nor disproved.

Formal system is a system that contains axioms, based on which new theorems can be defined. A number of mathematical approaches today are based on axiomatic definition. e.g. Probability theory. The number of axioms should be finite or there are ways which determine if the given statement is axiom or not. An axiomatic (formal) system is consistent if it doesn't contain the contradiction or if it is possible to prove that the statement and its negation are both true.

The Second incompleteness theorem is the extension of the first theorem which states that, any consistence axiomatic system, where certain elementary arithmetic operations are carried out, does not demonstrate its own consistency. Or in other words, if a formal system forms its own consistency then the system itself is inconsistent.

One of the interesting questions that always arose in my mind is that- "Is the universe analog or digital?" Is it only the outcome of 0 or 1 ? In other words, are there any possibilities that anything exist and do not exist at the same time? Godel's theorem could  answer them to some extent. 

Grandi's Series

After the name of Italian monk, priest, philosopher, mathematician and engineer Luigi Guido Grandi (1671-1742), the infinite series $1 - 1 + 1 - 1 \cdots \infty$, is called the Grandi's Series. It is a divergent series, meaning that if we take the partial sums of the series, the limit over the partial sum doesn't exist. Let us say the partial sum is

$S_1 =1 \\
 S_2= 1-1=0 \\
 S_3 = 1-1+1=1 \\
 S_4=1-1+1-1 =0\\
\vdots\\
S_n=\displaystyle\sum_{i=0}^n (-1)^i = 0 \;(\text {if $n$ is odd})\\
\quad\quad\quad \quad\quad\quad= 1 \;(\text {if $n$ is even})\\
\vdots\\
S_\infty=\displaystyle\sum_{i=0}^\infty (-1)^i$

Clearly, it is very difficult to conclude the value of the series. Grandi, however concluded that the sum of the series is $\frac{1}{2}$.

Consider the series,

$ S= 1 -1 + 1 -1 \cdots \infty, $ then

$1-S=1- 1 -1 + 1 -1 \cdots \infty = S  \Rightarrow 2S=1 \Rightarrow S=\frac{1}{2}. $

If we take the partial sum of the series and take the mean over it, then we obtain $\frac{1}{2}$. Such summation of infinite series is called Cesaro sum, after the name of another Italian mathematician Ernesto Cesaro (1859 - 1906).

Take the series $1+2+4+8+\cdots \infty$, What is the sum of the series?
Using the similar approach , Say
$c_1 = 1+2+4+8+\cdots \infty \\
\quad= 1+ 2(1+2+4+\cdots \infty)\\
\quad= 1+ 2c_1\\
\Rightarrow c_1-2c_1=1 \Rightarrow c_1=-1.$

These geometric series are quite trickier and gives the values way beyond expectation. Another such series, which we can obtain in similar way using Grandi's series is
$c_2= 1 -2 +3 -4 +5 -6 + \cdots \infty\\
c_2=\quad \;\;1 -2 +3 -4 +5 -6 + \cdots \infty\\
\underline {\hspace{8cm}} \\
2c_2=1-1+1-1+1-1+\cdots \infty\\
\Rightarrow 2c_2=\frac{1}{2}\\
\Rightarrow c_2= \frac{1}{4}.$

One of the interesting results which Indian mathematician Srinivasa Ramanujan (1887-1920) found in his earlier works is the sum of the series,
$c_3=1+2+3+4+\cdots \infty =\frac{-1}{12}$

We can express $c_3 - c_2$ as
$c_3=1+2+3+4+5+6+\cdots \infty \\
  c_2=1-2+3-4+5-6+\cdots \infty\\
-  \;\;  -\;\; +\;\;- \;\; +\;\; \; -\;\; \; +\;\;\;- \cdots \\
\underline {\hspace{8cm}}\\
 c_3-c_2=4+8+12+\cdots \infty\\
\Rightarrow c_3-c_2=4(1+2+3+\cdots \infty)\\
\Rightarrow c_3-c_2=4c_3\\
\Rightarrow-3c_3=c_2\\
\Rightarrow c_3=\frac{-c_2}{3}=\frac{-1}{12},\quad \text{as}\; c_2=\frac{1}{4} $