Compute summation of modules expression?In particular, what I want to look at is the sum...

MISA6zh

MISA6zh

Answered

2022-11-17

Compute summation of modules expression?
In particular, what I want to look at is the sum
k = 1 n ( p k ( mod q ) )
where p , q Z 1 can be assumed to be coprime but it would be best if solved in the fullest generality. In the above expression, n is a variable, p,q are fixed, and a(modb) means taking the representative set { 0 , 1 , 2 , . . . , b 1 }. For example, 7 ( mod 3 ) = 1 is the only value we agree upon and 7 ( mod 3 ) 2.
The problem with this is that the list of representatives are permuted by p and hence the methods presented in the initial link are no longer valid.
It would be nice if we can come up with a closed form, but a really tight upper bound of the expression also works.

Answer & Explanation

Patrick Arnold

Patrick Arnold

Expert

2022-11-18Added 21 answers

Step 1
Let the sequence 1p,2p,3p,...qp,...,np. This shows that there are some multiples of qp, or 1qp,2qp,...,rqp. Thus we can assume r be the largest number such that r q < n. We can calculate that the number of cycles are n q . The remaining numbers are rqmodn.
We assume that for some integer m 1 , m 2 , such that m 1 p k ( mod q ) and m 2 p k ( mod q ). By adding up the two equations we have ( m 1 m 2 ) p 0 ( mod q ). We know, for every prime p,q, they are coprime. This means that only for | m 1 m 2 | | q. This tells us that in the sets of number 0p,1p,2p,3p,...qp, the modulo of the numbers are always different.
As we take the modulo when 0 r q, thus for every set of numbers 1p,2p,...,qp, their modulo is 0 , 1 , 2 , . . . , q 1, the total sum of them is
n q i = 1 q 1 i = 1 2 n q q ( q 1 )
Step 2
Case 1: p < q, thus we can roughly calculate the modulo by hand.
Case 2: p > q, we can assume p = q + a for some integer a, we can also know that most of the value of a is even as mostly of the prime numbers are odd. Then, for some integer α, α p = α ( q + a ) = α q + α a α a ( mod q ). Thus the entire summation can be evaluated as follow:
k = 1 n p k mod q 1 2 n q q ( q 1 ) + i = 1 n r q i a mod q = 1 2 r ( q 2 q ) + i = 1 n r q i a mod q
Feel free to leave a comment if there're mistakes.

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