For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample ( i. e., a natural number n for which the Mertens function M ( n ) equals or exceeds the square root of n ) is known: all numbers less than 10 < sup > 14 </ sup > have the Mertens property, and the smallest number which does not have this property is only known to be less than the exponential of 1. 59 × 10 < sup > 40 </ sup >, which is approximately 10 to the power 4. 3 × 10 < sup > 39 </ sup >.

Given 40, the Mertens function returns 0.

The graph shows the Mertens function M ( n ) and the square roots ±√ n for n ≤ 10000.

In mathematics, the Mertens conjecture is the incorrect statement that the Mertens function M ( n ) is bounded by √ n, which implies the Riemann hypothesis.

In number theory, if we define the Mertens function as

where μ ( k ) is the Möbius function, then the Mertens conjecture is that for all n > 1,

At 31, the Mertens function sets a new low of-4, a value which is not subceded until 110.

Given 39, the Mertens function returns 0.

Given 58, the Mertens function returns 0.

Given 65, the Mertens function returns 0.

Given 93 the Mertens function returns 0.

At ninety-five, the Mertens function sets a new high of 2, after being below 0 for most of the numbers from 3 to 92.

Given 101, the Mertens function returns 0.

Following the prime quadruplet ( 101, 103, 107, 109 ), at 110, the Mertens function reaches a low of − 5.

At 114, the Mertens function sets a new low of-6, a record that stands until 197.

Given 150, the Mertens function returns 0.

Mertens function to n = 10, 000

Mertens function to n = 10, 000, 000

In number theory, the Mertens function is defined for all positive integers n as

Halle accepted Cantor's suggestion that it be offered to Dedekind, Heinrich M. Weber and Franz Mertens, in that order, but each declined the chair after being offered it.

After computing these values Mertens conjectured that the modulus of M ( n ) is always bounded by √ n.

Because the Möbius function only takes the values − 1, 0, and + 1, the Mertens function moves slowly and there is no n such that | M ( n )| > n. The Mertens conjecture went further, stating that there would be no n where the absolute value of the Mertens function exceeds the square root of n. The Mertens conjecture was proven false in 1985 by Andrew Odlyzko and Herman te Riele.

where M is the Meissel – Mertens constant.

A translation into English has been made by Silvanus P. Thompson (" Epistle of Peter Peregrinus of Maricourt, to Sygerus of Foucaucourt, Soldier, concerning the Magnet ", Chiswick Press, 1902 ); by Brother Arnold Charles Mertens (" The Letter of Petrus Peregrinus on the Magnet, A. D. 1269 ", with introductory note by Brother Potamian M. F. O ’ Reilly, York, 1904 ); and H. D. Harradon, (“ Some Early Contributions to the History of Geomagnetism-I ,” in Terrestrial Magnetism and Atmospheric Electricity Journal of Geophysical Research 48, 3-17 pp. 6 – 17 ).

The Mertens conjecture concerning its growth, conjecturing it bounded by x < sup > 1 / 2 </ sup >, which would have implied the Riemann hypothesis, is now known to be false ( Odlyzko and te Riele, 1985 ).

No analog of the Skewes number ( an upper bound on the first natural number x for which π ( x ) > li ( x )) is known in the case of Mertens ' 2nd and 3rd theorems.

The Meissel – Mertens constant is analogous to the Euler – Mascheroni constant, but the harmonic series sum in its definition is only over the primes rather than over all integers and the logarithm is taken twice, not just once.

Mertens is a town located in Hill County in Central Texas.

Mertens is located at ( 32. 058545 ,-96. 894887 ).

The Town of Mertens is served by the Frost Independent School District.

One of the most famous is César Franck but Henri Vieuxtemps, Eugène Ysaÿe, Guillaume Lekeu and Wim Mertens are also noteworthy.

The current Presiding Officer of the Municipal Council is Werner Mertens ( CD & V ).

331 is a prime number, cuban prime, sum of five consecutive primes ( 59 + 61 + 67 + 71 + 73 ), centered pentagonal number, centered hexagonal number, and Mertens function returns 0.

2 < sup > 2 </ sup > × 7 × 13, tetrahedral number, sum of twelve consecutive primes ( 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 ), Mertens function returns 0, nontotient, Harshad number.

401 prime number, tetranacci number, sum of seven consecutive primes ( 43 + 47 + 53 + 59 + 61 + 67 + 71 ), sum of nine consecutive primes ( 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 ), Chen prime, Eisenstein prime with no imaginary part, Mertens function returns 0, member of the Mian – Chowla sequence.

424 = 2 < sup > 3 </ sup > × 53, sum of ten consecutive primes ( 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 ), Mertens function returns 0, refactorable number, self number

607 prime number, sum of three consecutive primes ( 197 + 199 + 211 ), Mertens function ( 607 ) = 0, balanced prime, strictly non-palindromic number

5 × 127, sum of nine consecutive primes ( 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 ), Mertens function ( 635 )

2 < sup > 2 </ sup > × 3 × 53, sum of ten consecutive primes ( 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 ), Smith number, Mertens function ( 636 )

659 prime number, Sophie Germain prime, sum of seven consecutive primes ( 79 + 83 + 89 + 97 + 101 + 103 + 107 ), Chen prime, Mertens function sets new low of − 10 which stands until 661, highly cototient number, Eisenstein prime with no imaginary part, strictly non-palindromic number

661 prime number, sum of three consecutive primes ( 211 + 223 + 227 ), Mertens function sets new low of − 11 which stands until 665, star number