The Mertens function is defined as
where is the Möbius function. The function was named in honor of F.C.J. Mertens. Franz Carol Joseph Mertens was born on March 20th, 1840 in Sroda Prussia (now Środa Wielkopolska, Poland). He died on March 5th, 1927 in Vienna, Austria. The history of the attemps to proof the Riemann hypothesis started in 1885. Still in the records of the French Academy of Sciences, on July 13th, 1885, there is a note presented by Charles Hermite (the member of Academy) and written by a dutch mathematician Thomas Stjeltjes. He claims to have demonstrated the Riemann hypothese on one small page! The proof appeared false and Hermite explained why  . Stjeltjes never published the proof of the Riemann hypothesis. In 1897, Mertens based on empirical evidence claimed , named for him the “Mertens conjecture”  . In 1985, Mertens conjecture was disproved by te Riele and Odlyzko using a high-speed computer  . There are several popular books about the Riemann hypothesis. For the list of those books, we can find in the Google  . In  , we have the history of the zeta function of Riemann. Throughout this article, we will use letters: k, n for natural numbers, p for prime numbers and x for real numbers, and also we assume . The function . We will use Mertens function also for real numbers . is defined by
The estimation of Mertens function is important for the number theory by the theorem proved in 1912 by J. E. Littlewood  (p. 261).
Theorem: The statement
for every is equivalent to the Riemann hypothesis.
2. Some Properties of the Mertens Function
First we recall formula 
Next, we give the new formula involving Mertens function and Chebychev function
 (p. 107).
We shall prove the formula which was given by Mertens himself  , without a proof, (proposition 2.)
Formula is of the form
We state one of generalized Möbius inversion formulas  (p. 405) in the following form: let be a function for and
and reciprocally (vice versa).
Applying the Möbius formula as above to proposition 1 we get
On the other hand we have
Finally we have
This completes the proof.
Notice. The formulas used in the paper are some kind of identities. They follow from the properties of Mertens and Chebyshev functions.
3. The Estimation of Mertens Function
From proposition 1 we have
Because for all , where
we replaced by something greater, i.e. by and we get
for all ,  p. 318,
Notice. We use the symbol of sigma from 1 to infinity but the number of summand different from zero is always finite.
Next note that on the right hand side of above formula if then the corresponding summands
We know  , p. 319
(There are at most nonzero terms in last sum).
Finally, according to
From the definition of big “O” notation we have for all where .
Thus , i.e.
is the sufficient condition for the approval of Riemann hypothesis.
In  chapter 12.1, we can find more about that.
The estimation of the Mertens function is in the form as in theorem of the Lttlewood  . The result means the Riemann hypothesis is the theorem. The future problem is to find an exact formula for an imaginary part of the zeros of the Riemann zeta function.