All Questions
14
questions
0
votes
0
answers
19
views
Estimate the order of restricted number partitions
There are $k$ integers $m_l,1\leq l\leq k
$(maybe negetive), satisfiying $|m_l|\leq M$ and $\sum_l m_l=s$.
I want to get an order estimate of the number of solutions for $k, M$ when fixing $s$.
I came ...
- 91
2
votes
2
answers
81
views
Show that series converges by estimating number of partitions into distinct parts
I need some help with solving the following problem: Let $Q(n)$ be the number of partitions of $n$ into distinct parts. Show that $$\sum_{n=1}^\infty\frac{Q(n)}{2^n}$$ is convergent by estimating $Q(n)...
- 155
2
votes
0
answers
95
views
Power series of the form $r \bmod q$
I am trying exercises of Apostol Introduction to analytic number theory and I am struck on this problem of chapter partitions on page 14.
I am struck in part (b) as I have no idea on how to deal with ...
user775699
1
vote
1
answer
124
views
Question 14.12 Tom Apostol Introduction to Analytic number theory
I am studying Ch -14 from Apostol's book and could not solve this particular problem.
It's image:
I am unable to Solve 12(a) (I have done (b) ).
As q is prime, so (n, q) =1 or q| n and 11 (b) will be ...
user775699
1
vote
1
answer
62
views
Deriving a partition identity from some given identities
I am trying questions from Apostol Introduction to ANT of Chapter partitions and need help in deducing this identity.
Problem is question 6(a) which will use some information from 2 and 5(b).
...
user775699
4
votes
1
answer
234
views
2 identities of Shanks from Apostol's Book
I am trying exercises of Ch-14 partitions from Tom Apostol Introduction to ANT and unable to Solve (a) part of Question 5.
5. If $x\ne 1$ let $Q_0(x)=1$ and for $n\ge 1$ define
$$ Q_n(x) = \prod_{r=1}^...
user775699
4
votes
1
answer
199
views
How analytic continuation allows for proof of these 2 theorems in theory of Partitions
Consider these 2 theorems in textbook apsotol introduction to analytic number theory.
1st is generating functions for partitions
I have self studied text and need help in verifying the argument of ...
user775699
2
votes
1
answer
93
views
2 questions related to generating function of partition function in number theory
I am self studying chapter partitions (chapter number-14) from Apostol Introduction to analytic number theory.
I had studied that chapter earlier also and had questions but as I don't have anyone to ...
user775699
1
vote
1
answer
235
views
Number of unordered factorizations of a non-square-free positive integer
I recently discovered that the number of multiplicative partitions of some integer $n$ with $i$ prime factors is given by the Bell number $B_i$, provided that $n$ is a square-free integer. So, is ...
- 1,611
3
votes
1
answer
81
views
In how many ways can I express a positive integer as a sum of elements in a subset of $\mathbb Z^+$?
Let $S\subseteq \mathbb Z^+$ be set of positive integers. Given $n\in\mathbb Z^+$, how can I find the number of ways in which we can express $n$ as a sum of elements in $S$? ($S$ can be infinite.)
$$
...
- 7,534
2
votes
1
answer
171
views
Doubt in partition function generated by reciprocal of generating function of p(n).
While studying chapter partitions from Apostol introduction to analytic number theory I have a doubt on page number 311 .
Apostol defines inverse of partition function $\prod_{m=1}^{\infty} 1 - x^m $ ...
user775699
1
vote
0
answers
62
views
Waring problem generalizations and theta-function
My question is twofold:
Can the Waring problem be expressed with the Jacobi theta function
or some analog (as is the case for $k=2$) for general $k$? Say for $k=4$
or $k=6$, are these able to be ...
- 33
6
votes
1
answer
228
views
Recently proposed problem by George Andrews on partitions in Mathstudent Journal (India)
Show that the number of parts having odd multiplicities in all partitions of $n$ is equal to difference between the number of odd parts in all partitions of $n$ and the number of even parts in all ...
8
votes
2
answers
249
views
Minimizing over partitions $f(\lambda) = \sum \limits_{i = 1}^N |\lambda_i|^4/(\sum \limits_{i = 1}^N |\lambda_i|^2)^2$
I'm trying to characterize the behavior of the the quantity:
$$A = \frac{\sum \limits_{i = 1}^N x_i^4}{(\sum \limits_{i = 1}^N x_i^2)^2},$$
subject to the constraints that
$$ \sum \limits_{i = 1}^N ...
- 664