All Questions
Tagged with integer-partitions analytic-number-theory
9
questions with no upvoted or accepted answers
6
votes
0
answers
181
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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....
- 13.9k
4
votes
0
answers
229
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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 ...
- 727
2
votes
0
answers
95
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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
2
votes
0
answers
286
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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 $...
- 2,013
1
vote
0
answers
53
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Exercise I.0.9 from Tenenbaum's "Introduction to analytic and probabilistic number theory"
I'm trying to solve all the exercises from Tenenbaum's book but am unfortunately stuck on problem 9 of the very first ("tools") chapter. The problem is supposed to be an application of the ...
- 11
1
vote
1
answer
124
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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
0
answers
62
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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
0
votes
0
answers
19
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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
0
votes
0
answers
53
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Clarification of the proof of Euler's identity regarding the generating function for partitions.
In reference to this question which I asked here couple of days back but didn't get any answer I am posting this question to clarify whether we can able to extend Euler's identity regarding the ...
- 2,013