Question

This code takes an extremely long time to run (more than 10 minutes). Is there any way in which I can optimize it so that it finishes in less than one minute?

clear all;
for i = 1:1000000
    harmonicsum = 0;
    lhs = 0;
    for j = 1:i
        % compute harmonic sum
        harmonicsum = harmonicsum + 1/j;
        % find sum of factors
        if (mod(i,j)==0)
            lhs = lhs + j;
        end
    end
    %define right hand side (rhs) of Riemann Hypothesis
    rhs = harmonicsum + log(harmonicsum) * exp(harmonicsum);

    if lhs > rhs
        disp('Hypothesis violated')
    end
end

Answer 1

@b3 has a great vectorization of rhs.

One typo though, needs to use times and not mtimes:

harmonicsum = cumsum(1 ./ (1:1e6));
rhs = harmonicsum + log(harmonicsum) .* exp(harmonicsum);

For lhs, I propose the following, loosely based on Eratosthenes’ Sieve:

lhs = 1 + [1:1e6];
lhs(1) = 1;
for iii = 2:numel(lhs)/2
    lhs(2*iii:iii:end) = lhs(2*iii:iii:end) + iii;
end;

Execution time is just 2.45 seconds (for this half of the problem). Total including calculation of rhs and find is under 3 seconds.

I’m currently running the other version to make sure that the results are equal.

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EDIT: found a bug with lhs(1) and special-cased it (it is a special case, the only natural number where 1 and N aren’t distinct factors)