quantaum mathematics Chapter 8 uses of helium function

Chapter 8 Uses of helium Function

there is a lot of uses of helium function in solving prime numbers

1. first use is that it could be use to find the total number of distinct prime factors a number has using euler's totient function.
   

   
2. second use is about finding the number of primes between x(x+1) and (x+1)(x+2), i had found that number of primes between the the given range is equal to h(x(x+1)). it could be further be generalized to x(x+1)+1 and (x+1)(x+2)+1 because the the single value of helium function is almost constant or variation is very slow.
3. third use is in measuring the randomness of prime numbers, it could also be generalized for natural numbers.
   

randomness of prime graph is

it is interesting to note that natural number upto 12 have zero randomness. and the first randomness is found in natural number 13 of unit randomness also unit of randomness is chanchal which itself means randomness.

you can see the variation of blue line and red line .
4. helium function could be used to find the group number of primes.
h(n)=2h+1  n belongs to h group of prime if n is  replaced by 10^m then it is very useful.