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Problem A. 385. (November 2005)

A. 385. Find all sequences a1,a2,... of non-negative numbers, such that the sum S=\sum_{k=1}^\infty a_k is finite and \sum_{k=1}^\infty a_{kn} = \frac{S}{n^\alpha} for all positive integers n.

(5 pont)

Deadline expired on December 15, 2005.


Sketch of solution. We use the fact that the limit of the product \prod_p\left(1-\frac1{p^\alpha}\right) (running on all primes) is 0 for \alpha\le1 and is positive for \alpha>1.

Consider the first n primes p1,...,pn and sieve out the indices which are thie multilpies. We obtain

\sum_{p_1,\dots,p_n\not| k} a_k = 
\prod_{k=1}^n\left(1-\frac1{p_k^\alpha}\right) \cdot S.

If n\to\infty then the left-hand side converges to a1 because all other terms are sieved out.

If \alpha\le1 then the right-hand side converges to 0, therefore a1=0. Taking a positive integer l and repeating the same procedure for the subsequence akl we obtain al=0.

If \alpha>1 then the right hand side has a positive limit c_\alpha\cdot S. In this case a_l=\frac{c_\alpha S}{l^\alpha}.

Summarizing the results: for \alpha\le1 the only solution is the constant 0. For \alpha>1, the solution is a_k=\frac{c}{k^\alpha}.


Statistics:

9 students sent a solution.
5 points:Erdélyi Márton, Nagy 224 Csaba, Paulin Roland.
4 points:Gyenizse Gergő, Hujter Bálint, Kónya 495 Gábor.
3 points:1 student.
2 points:2 students.

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