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Let $K$ be an algebraic number field and let $I$ be an ideal in $O_K$ (the ring of integers).

Denote by $d(I)$ the number of ideals that divide $I$. So if $I= \prod_{i=1}^k p_i^{e_i}$ is the decomposition to prime ideals, then $d(I) = (e_1+1)...(e_k+1)$.

I am looking for references for known bounds of $d(I)$ in terms of $N(I)$ for the general case or for the case $I$ is principal ideal.

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It's certainly bounded above by the usual divisor function applied to $N(I)$. You will have equality exactly when each of the $p_i$ are totally split in $K$. Asymptotics for the usual divisor function can be found on its Wikipedia page. –  Kevin Ventullo Jul 16 '13 at 22:14
    
Thank you, Kevin. –  colge Jul 17 '13 at 15:35
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