Mertens function : new formulation

[Gaussianheart]

Is this identity true?

Look at attachment

Thank you.

 

[Gaussianheart]

Example :
o=25

M(50) = mu(27)+mu(29)+mu(31)+...+mu(49)
= 0+(-1)+(-1)+...+0= -3

mu(n) is Mobius function

 

[Gaussianheart]

It is true!
People checked it in other fora...
So thanks for reading the post.

 

[RamaWolf]

The traditional definition: M(n) = mu(1)+mu(2)+...+mu(n)

for n=8 we have M(8) = 1+(-1)+(-1)+0+(-1)+1+(-1)+0 = -2

Your formula Gh(8) = mu(4+2)+mu(4+4) = mu(6)+mu(8) = 1

in contradiction to the traditional formula

 

[Gaussianheart]

The traditional definition: M(n) = mu(1)+mu(2)+...+mu(n)

for n=8 we have M(8) = 1+(-1)+(-1)+0+(-1)+1+(-1)+0 = -2

Your formula Gh(8) = mu(4+2)+mu(4+4) = mu(6)+mu(8) = 1

in contradiction to the traditional formula

n=8=2*4 an 4 is not odd

Read the condition : o must be odd >=3 then you can compute M(2*o)

My formula holds. Someone in another forum just proved it.
I have a proof but it is little bit long.

Thank you for your comment

 

[RamaWolf]

With n=2*o, (o odd) it's OK

For my investigation I used ARIBAS (Windows version) and I programmed a function 'SmallMoebiusMu(n)' (small because n must not exceed 2**32) and with this function, I compared my function 'SmallMertensNumber(n)' (traditional definition) to the function 'Gaussianheart(n)' (your formulation) and for n=2,6,10,14,18,...,402 I found equal results.

SmallMoeniusMu uses the built-in ARIBAS function 'factor16' and 'prime32test'.

Regards from Germany

 

[Gaussianheart]

The formula is correct!
Good for me!