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In this paper , we prove two formulas involving Mertens and Chebyshev functions. The first formula was done by Mertens himself without a proof. The second formula is a new one. Using these formulas , we estimate the Mertens function in such manner that we obtain a sufficient condition to approve the Riemann hypothesis.

The Mertens function M ( n ) is defined as

M ( n ) = ∑ k ≤ n μ ( k ) (1)

where μ ( k ) 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 [

M ( 0 ) = 0 , (2)

M ( [ x ] ) = ∑ k ≤ [ x ] μ ( k ) . (3)

The estimation of Mertens function is important for the number theory by the theorem proved in 1912 by J. E. Littlewood [

Theorem: The statement

M ( x ) = O ( x 1 2 + ϵ ) (4)

for every ϵ > 0 is equivalent to the Riemann hypothesis.

First we recall formula [

∑ k ≤ x M ( x k ) = 1 . (5)

Next, we give the new formula involving Mertens function and Chebychev function

ψ ( x ) = ∑ p m ≤ x log ( p ) . (6)

Proposition 1.

M ( x ) log ( x ) = ∑ k ≤ x μ ( k ) ( log ( x k ) − ψ ( x k ) ) . (7)

∑ k ≤ x μ ( k ) ( log ( x k ) − ψ ( x k ) ) = log ( x ) ∑ k ≤ x μ ( k ) − ∑ k ≤ x μ ( k ) log ( k ) (8)

− ∑ k ≤ x μ ( k ) ψ ( x k ) = M ( x ) log ( x ) − ∑ k ≤ x μ ( k ) log ( k ) − ∑ k ≤ x μ ( k ) ψ ( x k ) (9)

= M ( x ) log ( x ) (10)

because

∑ k ≤ x μ ( k ) log ( k ) = − ∑ k ≤ x μ ( k ) ψ ( x k ) . (11)

[

We shall prove the formula which was given by Mertens himself [

Formula is of the form

Proposition 2.

ψ ( x ) = ∑ k ≤ x M ( x k ) log ( k ) . (12)

We state one of generalized Möbius inversion formulas [

g ( x ) = ∑ k ≤ x f ( x k ) . (13)

Then for x ≥ 1

f ( x ) = ∑ k ≤ x μ ( k ) g ( x k ) , (14)

and reciprocally (vice versa).

Applying the Möbius formula as above to proposition 1 we get

∑ k ≤ x M ( x k ) log ( x k ) = log ( x ) − ψ ( x ) . (15)

On the other hand we have

∑ k ≤ x M ( x k ) log ( x k ) = ∑ k ≤ x M ( x k ) log ( x ) − ∑ k ≤ x M ( x k ) log ( k ) (16)

= log ( x ) ∑ k ≤ x M ( x k ) − ∑ k ≤ x M ( x k ) log ( k ) (17)

= log ( x ) − ∑ k ≤ x M ( x k ) log ( k ) . (18)

Finally we have

log ( x ) − ψ ( x ) = log ( x ) − ∑ k ≤ x M ( x k ) log ( k ) , (19)

so

ψ ( x ) = ∑ k ≤ x M ( x k ) log ( k ) . (20)

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.

From proposition 1 we have

M ( x ) log ( x ) = ∑ k ≤ x μ ( k ) log ( x k ) − ∑ k ≤ x μ ( k ) ψ ( x k ) (21)

| M ( x ) log ( x ) | = | ∑ k ≤ x μ ( k ) log ( x k ) − ∑ k ≤ x μ ( k ) ψ ( x k ) | (22)

= | ∑ k ≤ x μ ( k ) ( log ( x k ) − ψ ( x k ) ) | (23)

≤ ∑ k ≤ x | log ( x k ) − ψ ( x k ) | .

Because log x ≤ θ ( x ) for all x ≥ 1 , where

θ ( x ) = ∑ p ≤ x log ( p ) (24)

we replaced log ( x k ) by something greater, i.e. by θ ( x k ) and we get

∑ k ≤ x | log ( x k ) − ψ ( x k ) | ≤ ∑ k ≤ x | θ ( x k ) − ψ ( x k ) | .

We have

ψ ( x ) = ∑ m = 1 ∞ θ ( x 1 / m ) (25)

for all x ≥ 1 , [

and

ψ ( x k ) = ∑ m = 1 ∞ θ ( ( x k ) 1 / m ) . (26)

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 ( x k ) 1 / m < 2 then the corresponding summands

θ ( ( x k ) 1 / m ) = 0. (27)

Let

m = ( log ( x k ) ) / log ( 2 ) = ( log ( x ) − log ( k ) ) / log ( 2 ) = log ( x ) log ( 2 ) − log ( k ) log ( 2 ) . (28)

If m > log x log 2

then

θ ( ( x k ) 1 / m ) = 0. (29)

ψ ( x k ) = ∑ m = 1 ∞ θ ( ( x k ) 1 / m ) . (30)

ψ ( x k ) − θ ( x k ) = ∑ m = 2 ∞ θ ( ( x k ) 1 / m ) . (31)

We know [

θ ( x ) = O ( x log ( x ) ) (32)

and

ψ ( x k ) − θ ( x k ) = ∑ m = 2 ∞ O ( ( x k ) 1 / m log ( x k ) ) (33)

= O ( ( x k ) 1 / 2 log ( x k ) ) + ∑ 3 ≤ m ≤ log ( x ) / log ( 2 ) O ( ( x k ) 1 / 3 log 2 ( x k ) ) (34)

= O ( ( x k ) 1 / 2 log ( x k ) ) . (35)

(There are at most log ( x k ) nonzero terms in last sum).

Finally, according to

| M ( x ) log ( x ) |

≤ ∑ k ≤ x | θ ( x k ) − ψ ( x k ) | = ∑ k ≤ x ( ψ ( x k ) − θ ( x k ) ) (36)

= ∑ k ≤ x O ( ( x k ) 1 / 2 log ( x k ) ) = O ( x 1 / 2 log ( x ) ) , (37)

we obtain

| M ( x ) | log ( x ) = O ( x 1 / 2 log ( x ) ) . (38)

From the definition of big “O” notation we have | M ( x ) | log ( x ) ≤ K x 1 2 log ( x ) for all x ≥ 1 where K > 0 .

Thus | M ( x ) | ≤ K x 1 2 , x ≥ 1 i.e.

M ( x ) = O ( x 1 2 ) . (39)

The result

M ( x ) = O ( x 1 2 ) (40)

is the sufficient condition for the approval of Riemann hypothesis.

In [

The estimation of the Mertens function M ( x ) is in the form as in theorem of the Lttlewood [

The author declares no conflict of interest regarding the publication of this paper.

Czopik, J. (2019) The Estimation of the Mertens Function. Advances in Pure Mathematics, 9, 415-420. https://doi.org/10.4236/apm.2019.94019