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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 to this question when a series sum on $k$ containing this number and to get the summation's relation with $M$.

I have tried for plus all the integers on $M$, then it transforms to a traditional number partition problem for $s'=kM+s$ with restrict $0\leq m_l\leq 2M$, remove the restrict and by the result of Hardy and Ramanujan get the order is $e^{\sqrt{kM+s}}$. but it is not enough to solve my problem, I expect to have a polynomial bound for $k$ or $M$, do you have some methods to refine the result?

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    $\begingroup$ The number of positive integer solutions to $x_1+x_2+\dots+x_k=n$ is $\binom{n-1}{k-1}$. $\endgroup$
    – TravorLZH
    Commented Apr 6 at 17:38
  • $\begingroup$ Thank you, I forgot the $k$ can be fixed. By the way, the number of $p_n$ is $\sum_{k=1}^n\binom{n-1}{k-1}$? $\endgroup$
    – Trinifold
    Commented Apr 8 at 0:28

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