about Ramanujan's work

1. By
Dr T N Kavitha
Assistant Professor of Mathematics
SCSVMV

2. An Introduction to Ramanujan
Srinivasa Ramanujan (1887-1920) was an
Indian mathematician known for his
extraordinary contributions to mathematical
analysis, number theory, infinite series, and
continued fractions. Despite having very little
formal training in mathematics, Ramanujan
independently discovered a wealth of results
that have had a profound impact on various
branches of mathematics.

3. Here are a few key aspects of
his work:
• Infinite Series and Continued Fractions:
• Mock Theta Functions:
• Partition Function:
• Ramanujan Prime and Ramanujan-Hardy
Number:
• Modular Forms:
• Elliptic Integrals:

4. His brilliance & his contributions
Ramanujan's work was largely unrecognized
during his lifetime, but after his death,
mathematicians such as G. H. Hardy recognized
the brilliance of his contributions and worked to
ensure the dissemination and preservation of his
results. Ramanujan's notebooks, which contain
much of his unpublished work, continue to be a
source of inspiration and research for
mathematicians around the world.

5. Ramanujan Prime and
Ramanujan-Hardy Number:
Ramanujan made several interesting
discoveries related to prime numbers,
including the discovery of what is now
known as the "Ramanujan primes." He also
found an approximation for the number of
prime numbers less than a given value,
known as the Ramanujan-Hardy number.

6. Ramanujan-Hardy Number:
The Ramanujan-Hardy number is an approximation for
the number of prime numbers less than a given integer.
It is named after Ramanujan and the British
mathematician G. H. Hardy.
The formula for the Ramanujan-Hardy number H(x) is
given by:
H(x)=(6/ π2)​x
This formula provides an estimate of the number of
primes less than x and is particularly accurate for large
values of x.
The constant 6/π2 ​ is derived from the asymptotic
behavior of certain arithmetic functions.

7. Example
If you want to estimate the number of primes
less than 100 using the Ramanujan-Hardy
formula:
H(100) = (6/ π2)×100 ≈60.81
So, according to this approximation, there are
about 60 prime numbers less than 100.

8. Ramanujan-Hardy Number
(Taxicab Number):
Ramanujan primes and the Ramanujan-Hardy
number.1729 is called Ramanujan-Hardy number
because Ramanujan knew 1729 is the smallest
number that can be expressed as the sum of two
cubes in two different ways.
10³ + 9³ = 1729
12³ + 1³ = 1729

9. Similar Taxicab numbers
4104 = 163 + 23
= 153 + 93
87539319 = 2283 + 4233
= 1673 + 4363
= 2553 + 4143

10. Ramanujan Primes:
Ramanujan discovered a class of prime numbers,
which are now known as Ramanujan primes.
These primes, denoted as R_n, have
the property that they are the smallest prime
number that divides the coefficient of the nth
term in the expansion of certain modular forms.

11. Modular form of Ramanujan
primes
The specific modular form associated with
Ramanujan primes is related to the partition
function, denoted as p(n).
The nth Ramanujan prime, R_n, is
associated with the coefficient of q^n in the
expansion of a particular modular form.
The precise definition involves a function
denoted as q, which is related to the modular
form.

12. The first few Ramanujan primes
The first few Ramanujan primes are as follows:
R₁ = 2
R₂ = 11
R₃ = 17
R₄ = 29
These primes have interesting connections to
number theory, modular forms, and the
distribution of prime numbers.

13. Ramanujan Primes
It's important to note that while Ramanujan's
formula gives a good approximation for the
distribution of primes, it's not an exact count and
serves as a heuristic for understanding the
behavior of prime numbers.
These concepts showcase Ramanujan's ability to
connect different areas of mathematics and
provide insightful results that continue to be
studied and appreciated by mathematicians.

14. A number of primes less than or
equal to x
A Ramanujan prime is a prime number that
satisfies a result proved by Srinivasa Ramanujan
relating to the prime counting function.
The n th Ramanujan prime is the least positive
integer Rₙ for which
π(x)—π(x/2) ≥ n ,∀ x ≥ Rₙ , n ≥ 1
where π(x) is the prime counting function
(number of primes less than or equal to x).
In other words, there are at least n primes
between x/2 and x whenever x≥Rₙ

15. 142857 is called Ramanujan’s
constant.
The complete multiplication table of 142857 can be found. Multiplication by 8 ,
9 & 19 in the below sample listing is worth noting. The first and last digit totals
to 7 , 4 and 5 respectively bringing the answer within the circle !!!!!!
This pattern is found for ALL multiples of 142857.
142857 x 1 = 142857
142857 x 2 = 285714
142857 x 3 = 428571
142857 x 4 = 571428
142857 x 5 = 714285
142857 x 6 = 857142
142857 x 8 = 1142856
142857 x 9 = 1285713
142857 x 19 = 2714283
142857 x 7 = 999999
And finally, 142857 can be represented as 1/7.

16. What are the Ramanujan prime
numbers?
Ramanujan primes Rn: a(n) is the smallest number
such that if x ≥ a(n), then pi(x) - pi(x/2) ≥ n, where
pi(x) is the number of primes ≤ x.
{2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127,
149, 151, 167, 179, 181, 227, 229, 233, 239, 241,
263, 269, 281, 307, 311, 347, 349, 367, 373, 401,
409, 419, 431, 433, ...}

17. Ramanujan primes of the second
kind:
Ramanujan primes of the second kind: a(n) is the
smallest prime such that if prime x≥ a(n), then
pi(x) - pi(x/2) ≥ n, where pi(x) is the number of
primes ≤ x.
{2, 3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103,
109, 131, 151, 157, 173, 181, 191, 229, 233, 239,
241, 251, 269, 271, 283, 311, 313, 349, 353, 373,
379, 409, 419, 421, 433, ...}

18. Ramanujan Paradox
Ramanujan showed that any big numbers
can be written as sum of not more than four
prime numbers
but we can add 2,3,5,7,11,13,17, etc. to form
a number and hence the number obtained is
sum of more than four prime numbers. Why
so?

19. The sum of not-more-than-four-prime-
numbers.
Any big number can be written as sum of not more than four
prime numbers
and
Any big number can only be written as sum of not more than four
prime numbers
The first statement says that while you might be able to write a
number as a sum of, say 20 prime numbers, but there will always
be another way to write this number using four or less prime
numbers. On the other hand, the second statement denies the fact
that alternate representations are possible.
As a concrete example, while you can always write 58 = 2 + 3 +
5 + 7 + 11 + 13 + 17, you can also write 58 as 53 + 5, which is
the sum of not-more-than-four-prime-numbers.

20. Ramanujan magic square is as
follows:

21. Ramanujan’s magic square.
Here is a simple method to find magic
squares for any given Date of Birth, similar
to Ramanujan’s magic square.
Let d = date, m=month, Y = first two digits of
year, y=second two digits of the year. Then
the date of birth magic squares can be
formed using below.

22. Let d = date, m=month, Y = first two digits of year,
y=second two digits of the year. Then the date of birth
magic squares can be formed using below.

23. Eg : Given Date of Birth is : 05–06–1968 ->value of
d=5, m=6, Y=19, y=68 in the above gives us the
following Magic square of sum = 98

24. Thank you