Proof that Harmonic numbers Hn = 1+1/2+1/3+...+1/n can not be calculated in constant number of arithmetic operations ?

Consider the basic arithmetical operations of addition, subtraction, multiplication and division.

Can the sum H_n = 1+1/2+1/3+..+1/n be expressed in number of these basic operations that does not depend on n ? Supposedly this can not be done but what is the proof ?

For example the sum:

S_n = 1+2+3+...+n with n-1 additions can be expressed compactly by Gauss formula as n*(n+1)/2 where the number of operations is three and (so it does not depend on n).

I need a similar thing for H_n, or a proof that this can not be done.