All Questions
Tagged with integer-partitions analytic-number-theory
21
questions
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
$$ \sum \limits_{i = 1}^N ...
6
votes
1
answer
228
views
Recently proposed problem by George Andrews on partitions in Mathstudent Journal (India)
Show that the number of parts having odd multiplicities in all partitions of $n$ is equal to difference between the number of odd parts in all partitions of $n$ and the number of even parts in all ...
6
votes
0
answers
181
views
Almost a prime number recurrence relation
For the number of partitions of n into prime parts $a(n)$ it holds
$$a(n)=\frac{1}{n}\sum_{k=1}^n q(k)a(n-k)\tag 1$$
where $q(n)$ the sum of all different prime factors of $n$.
Due to https://oeis....
4
votes
1
answer
234
views
2 identities of Shanks from Apostol's Book
I am trying exercises of Ch-14 partitions from Tom Apostol Introduction to ANT and unable to Solve (a) part of Question 5.
5. If $x\ne 1$ let $Q_0(x)=1$ and for $n\ge 1$ define
$$ Q_n(x) = \prod_{r=1}^...
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 ...
4
votes
0
answers
229
views
Newman's proof of the Asymptotic Formula for the Partition Function
I'm working on Donald J. Newman's proof that $p(n) \sim \frac{1}{4\sqrt{3}n}e^{\pi\sqrt{\frac{2n}{3}}}$, as found in Chapter II of his book Analytic Number Theory.
Here's what we have so far: the ...
3
votes
1
answer
169
views
How to extend Euler's identity regarding partition on the unit disk?
Theorem (Euler) $:$ For $|x|<1$ we have
$$\prod\limits_{m=1}^{\infty} \frac {1} {1-x^m} = \sum\limits_{n=0}^{\infty} p(n) x^n,$$ where $p(n)$ denotes the number of partitions of $n$ for $...
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.)
$$
...
2
votes
2
answers
81
views
Show that series converges by estimating number of partitions into distinct parts
I need some help with solving the following problem: Let $Q(n)$ be the number of partitions of $n$ into distinct parts. Show that $$\sum_{n=1}^\infty\frac{Q(n)}{2^n}$$ is convergent by estimating $Q(n)...
2
votes
1
answer
93
views
2 questions related to generating function of partition function in number theory
I am self studying chapter partitions (chapter number-14) from Apostol Introduction to analytic number theory.
I had studied that chapter earlier also and had questions but as I don't have anyone to ...
2
votes
0
answers
95
views
Power series of the form $r \bmod q$
I am trying exercises of Apostol Introduction to analytic number theory and I am struck on this problem of chapter partitions on page 14.
I am struck in part (b) as I have no idea on how to deal with ...
2
votes
1
answer
171
views
Doubt in partition function generated by reciprocal of generating function of p(n).
While studying chapter partitions from Apostol introduction to analytic number theory I have a doubt on page number 311 .
Apostol defines inverse of partition function $\prod_{m=1}^{\infty} 1 - x^m $ ...
2
votes
0
answers
286
views
Recurrence relation for partition function for pentagonal numbers.
I know the following theorems.
Theorem 1 $:$ For $|x|<1$ we have $$\prod\limits_{k=1}^{\infty} \frac {1} {1-x^k} = 1 + \sum\limits_{k=1}^{\infty} p(k)x^k.$$
Theorem 2 $:$ For $|x|<1$ we have $...
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
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).
...