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Summatory function of the Möbius function
In number theory, the Mertens function is defined for all positive integers n as M ( n ) = ∑ k = 1 n μ ( k ) , {\displaystyle M(n)=\sum _{k=1}^{n}\mu (k)
Mertens_function
= Sophie Germain prime, centered square number, Mertens function zero 1014 = 210-10, Mertens function zero, sum of the nontriangular numbers between successive
1000_(number)
Natural number
quadruplet 2093 – Mertens function zero 2095 – Mertens function zero 2096 – Mertens function zero 2097 – Mertens function zero 2099 – Mertens function zero, super-prime
2000_(number)
Disproved mathematical conjecture
In mathematics, the Mertens conjecture is the statement that the Mertens function M ( n ) {\displaystyle M(n)} is bounded by ± n {\displaystyle \pm {\sqrt
Mertens_conjecture
Multiplicative function in number theory
OEIS). In number theory another arithmetic function closely related to the Möbius function is the Mertens function, defined by M ( n ) = ∑ k = 1 n μ ( k )
Möbius_function
Natural number
a zero of Mertens function. 812 = 22 × 7 × 29. It is an admirable number, a pronic number, a balanced number, and a zero of Mertens function. 813 = 3 ×
800_(number)
Natural number
Eisenstein prime with no imaginary part, a tetranacci number, a zero of Mertens function, and a member of the Mian–Chowla sequence. It is the sum of seven consecutive
400_(number)
Natural number
consecutive primes (7 + 11 + 13 + 17 + 19 + 23 + 29 + 31). Given 150, the Mertens function returns 0. 150 is conjectured to be the only minimal difference greater
150_(number)
Polish-Austrian mathematician
Franz Mertens (20 March 1840 – 5 March 1927) (also known as Franciszek Mertens) was a German-Polish mathematician. He was born in Schroda in the Grand
Franz_Mertens
Surname list
Frank Mertens (born 1961), German keyboardist and composer Franz Mertens (1840–1927), German mathematician Mertens conjecture, Mertens function, Mertens' theorems
Mertens
Natural number
zero of Mertens function and a strictly non-palindromic number 608 = 25 × 19. It is a nontotient, a happy number, and a zero of the Mertens function. There
600_(number)
Natural number
hand, the first two integers that return 0 {\displaystyle 0} for the Mertens function (2 and 39) have a difference of 37, where their product (2 × 39) is
37_(number)
Natural number
Schröder–Hipparchus number, Mertens function(903) returns 0, little Schroeder number 904 = 23 × 113 or 113 × 8, refactorable number, Mertens function(904) returns 0
900_(number)
Natural number
zero of Mertens function. It is the sum of five consecutive primes (59 + 61 + 67 + 71 + 73). 332 = 22 × 83. It is a zero of Mertens function. 333 = 32
300_(number)
Natural number
– Mertens function zero 8011 – Mertens function zero, super-prime 8012 – Mertens function zero 8017 – Mertens function zero 8021 – Mertens function zero
8000_(number)
Natural number
362=2\times 181=\sigma _{2}(19)} : sum of squares of divisors of 19, Mertens function returns 0, nontotient, noncototient. 364 = 2 2 × 7 × 13 {\displaystyle
360_(number)
Natural number
the sum of the first three powers of 3 (31 + 32 + 33). Given 39, the Mertens function returns 0. 39 is the smallest natural number which has three partitions
39_(number)
Natural number
the Mertens function. There are 795 permutations of length 7 with 2 consecutive ascending pairs. 796 = 22 × 199. It is a zero of the Mertens function and
700_(number)
Natural number
thirty-second number to return 0 for the Mertens function M(n). Sloane, N. J. A. (ed.). "Sequence A002088 (Sum of totient function)". The On-Line Encyclopedia of
32_(number)
Natural number
as the sum of the cubes of the first three primes. Given 160, the Mertens function returns 0. 160 is the smallest number n with exactly 12 solutions to
160_(number)
Arithmetic function
^{-1}} -weighted summatory functions are related to the Mertens function, or weighted summatory functions of the Möbius function. In fact, we have that the
Liouville_function
Natural number
537 = 3 × 179. It is a Blum integer, a D-number, and a zero of the Mertens function. 538 = 2 × 269. It is a nontotient and an open meandric number. Other
500_(number)
Natural number
which spells a proper noun with multiple meanings. Given 159, the Mertens function returns 0. "Sloane's A003261 : Woodall (or Riesel) numbers". The On-Line
159_(number)
Conjecture on zeros of the zeta function
Riemann hypothesis is equivalent to this bound for the Möbius function μ and the Mertens function M derived in the same way from it. In other words, the Riemann
Riemann_hypothesis
Three results related to the density of prime numbers
analytic number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens. In the following, let
Mertens'_theorems
Square (0,1) matrix
signs). As a corollary of the disproof of the Mertens conjecture, it follows that the Mertens function changes sign, and is therefore zero, infinitely
Redheffer_matrix
Natural number
is an octagonal number. It is also a Cullen number. Given 65, the Mertens function returns 0. This number is the magic constant of a 5×5 normal magic
65_(number)
Natural number
it is also the fiftieth number to return 0 {\displaystyle 0} in the Mertens function. While the twenty-first prime number 73 is the largest member of Bhargava's
21_(number)
Natural number
the sum of the first twenty positive even numbers. a zero of the Mertens function and is sparsely totient. a pronic number. The least common multiple
420_(number)
Natural number
number. the lowest integer for which the Mertens function is greater than 1. (The lowest integer producing a Merten's value greater than that of 95 is 218)
95_(number)
Natural number
of those bases, it is a strong pseudoprime: 1, 12, 17, and 144. The Mertens function returns 0. 145 is a pentagonal number and a centered square number
145_(number)
Natural number
palindromic in any base between base 2 and base 161. Given 163, the Mertens function returns 0. It is the fourth prime with this property. The first three
163_(number)
Natural number
2 (111001112). 231 is the number of integer partitions of 16. The Mertens function of 231 returns 0. A US gallon is defined by being exactly 231 cubic
231_(number)
Natural number
sum of the first four hyperfactorials, including H(0). At 114, the Mertens function sets a new low of -6, a record that stands until 197. 114 is the smallest
114_(number)
Number of prime factors of a natural number
B_{1}\approx 0.26149721} is the Mertens constant and γ j {\displaystyle \gamma _{j}} are the Stieltjes constants. The function ω ( n ) {\displaystyle \omega
Prime_omega_function
Mathematical constant
The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as the Mertens constant, Kronecker's constant (after Leopold
Meissel–Mertens_constant
Natural number
natural number following 163 and preceding 165. 164 is a zero of the Mertens function. In base 10, 164 is the smallest number that can be expressed as a
164_(number)
Natural number
natural number following 218 and preceding 220. 219 is a happy number. Mertens function (219) = 4, a record high. There are 219 partially ordered sets on four
219_(number)
Natural number
composite number. It is a centered triangular number. Given 166, the Mertens function returns 0. 166 is a Smith number in base 10. 166 in Roman numerals
166_(number)
Natural number
Following the prime quadruplet (101, 103, 107, 109), at 110, the Mertens function reaches a low of −5. 110 is the sum of three consecutive squares, 110
110_(number)
Natural number
eight blank stamps into a single flat pile of stamps. 353 is a zero of Mertens Function. 353 is an index of a prime Lucas number. Sloane, N. J. A. (ed.). "Sequence
353_(number)
Natural number
[and] eighteen) is the natural number following 217 and preceding 219. Mertens function (218) = 3, a record high. 218 is nontotient and also noncototient.
218_(number)
In mathematics, a statement that has been proven
n for which the Mertens function M(n) equals or exceeds the square root of n) is known: all numbers less than 1014 have the Mertens property, and the
Theorem
Hilbert–Pólya conjecture Generalized Riemann hypothesis Mertens function, Mertens conjecture, Meissel–Mertens constant De Bruijn–Newman constant Dirichlet character
List_of_number_theory_topics
Formula for the sum of an arithmetic function
character. Other examples appear in the articles on the Mertens function and the von Mangoldt function. Perron's formula is a special case of the formula ∑
Perron's_formula
Characterization of how many integers are prime
∑ n ≤ x μ ( n ) {\displaystyle M(x)=\sum _{n\leq x}\mu (n)} is the Mertens function. Based on the tables by Anton Felkel and Jurij Vega, Adrien-Marie Legendre
Prime_number_theorem
Increasing sequence of reduced fractions
{3(|F_{n}|-1)}{2}}-n-\left\lceil {\frac {n}{2}}\right\rceil ,} The Mertens function can be expressed as a sum over Farey fractions as M ( n ) = − 1 + ∑
Farey_sequence
the disproof of Mertens conjecture given by Odlyzko and te Riele in their several decades old breakthrough paper Disproof of the Mertens Conjecture. In
Extremal orders of an arithmetic function
Extremal_orders_of_an_arithmetic_function
Integration by parts version of Abel's method for summation by parts
n ≤ x μ ( n ) {\displaystyle A(x)=M(x)=\sum _{n\leq x}\mu (n)} is Mertens function and 1 ζ ( s ) = ∑ n = 1 ∞ μ ( n ) n s = s ∫ 1 ∞ M ( u ) u 1 + s d u
Abel's_summation_formula
Belgian game theorist (1946–2012)
Jean-François Mertens (11 March 1946 – 17 July 2012) was a Belgian game theorist and mathematical economist. Mertens contributed to economic theory in
Jean-François_Mertens
Family of probability distributions
applications health economics meteorology and climatology fisheries Mertens function self-organized criticality Tweedie, M.C.K. (1984). "An index which
Tweedie_distribution
Mathematical operation on arithmetical functions
{\displaystyle M(x)} is the Mertens function and ω {\displaystyle \omega } is the distinct prime factor counting function from above. This expansion follows
Dirichlet_convolution
Number of integers coprime to and less than n
( x ) {\displaystyle \log _{e}(x)} . In number theory, Euler's totient function counts the positive integers up to a given integer n {\displaystyle n}
Euler's_totient_function
industry in the United States. Franciszek Mertens, mathematician known for Mertens function, Mertens conjecture, Mertens's theorems. Josef Hofmann, designer of
Timeline of Polish science and technology
Timeline_of_Polish_science_and_technology
Empirical law on the variance of species in a habitat
(SNPs) gene structures in number theory with sequential values of the Mertens function and also with the distribution of prime numbers from the eigenvalue
Taylor's_law
Mathematical operation
involving the Mertens function, or summatory function of the Moebius function, the prime zeta function and the prime-counting function, and the Riemann
Dirichlet_series_inversion
Arithmetic function related to the divisors of an integer
theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number
Divisor_function
Concept in game theory
deployed to extend this diagonal formula when the function f is no longer differentiable. Mertens goes back to the original formula and takes the derivative
Shapley_value
Canadian-American physicist and academic
"Randomness of Mobius coefficients and Brownian motion: growth of the Mertens function and the Riemann Hypothesis". Journal of Statistical Mechanics: Theory
André_LeClair
Difference between logarithm and harmonic series
of the divisor function. A formulation of the Riemann hypothesis. The third of Mertens' theorems.* The calculation of the Meissel–Mertens constant. Lower
Euler's_constant
Belgian-French writer and lawyer (1939–2025)
Belga (19 January 2025). "L'écrivain belge Pierre Mertens est décédé". DHnet (in French). Retrieved 19 January 2025. Pierre Mertens at ARLFFB (in French)
Pierre_Mertens
Madeleine Martin) is a human girl and the gender-swapped version of Finn Mertens. Cake (voiced by Roz Ryan) is Fionna's cat and Jake's gender-swapped equivalent
List of Adventure Time characters
List_of_Adventure_Time_characters
Function computable with bounded loops
In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all
Primitive_recursive_function
Concept in mathematics
(an)n≥0 and (bn)n≥0 be real or complex sequences. It was proved by Franz Mertens that, if the series ∑ n = 0 ∞ a n {\textstyle \sum _{n=0}^{\infty }a_{n}}
Cauchy_product
Problem in computer science
in his index. Davis 1958, pp. vii–viii. Davis 1958, pp. 70–71. Moore & Mertens 2011, pp. 236–237. Strachey, C. (1 January 1965). "An impossible program"
Halting_problem
Decision rule of maximizing utility
abstract social choice function, relative utilitarianism has been analyzed by Cao (1982), Dhillon (1998), Karni (1998), Dhillon and Mertens (1999), Segal (2000)
Utilitarian_rule
Financial tool for tracking orders by buyers and sellers
specialist matching orders for the specific item. In his work, Jean-François Mertens extends this and constructs an order matching mechanism that works across
Order_book
Dutch optician
counterfeiter) Known for Possible inventor of the microscope and the telescope (posthumous claim) Parents Hans Mertens (father) Mayken Provoost Bacher (mother)
Zacharias_Janssen
Open problem on 3x+1 and x/2 functions
positive integers, as in the case of the disproven Pólya conjecture and Mertens conjecture. However, such verifications may have other implications. Certain
Collatz_conjecture
Mathematical proportionality to a square
communications network grows quadratically as a function of its number of users. Exponential growth Moore, Cristopher; Mertens, Stephan (2011), The Nature of Computation
Quadratic_growth
Comedy Habib Rhapsody [de] Michael Baumann [de] Vedat Erincin, Thorsten Merten [de], Burak Yiğit [de], Klaus Manchen [de] Drama a.k.a. Willkommen bei Habib
List of German films of the 2010s
List_of_German_films_of_the_2010s
2005 aviation accident in Greece
where it was due to arrive at 10:45. In command was Captain Hans-Jürgen Merten, a 59-year-old German contract pilot hired by Helios for holiday flights
Helios_Airways_Flight_522
Algorithm in computational number theory
algorithm was its use by Andrew Odlyzko and Herman te Riele in disproving the Mertens conjecture. The LLL algorithm has found numerous other applications in
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Number divisible only by 1 and itself
{\displaystyle x} . The growth rate of this sum is described more precisely by Mertens' second theorem. For comparison, the sum 1 1 2 + 1 2 2 + 1 3 2 + ⋯ + 1
Prime_number
Documentary Solo für Georg Jens-Peter Behrend [de] Helmut Baumann [de], Michaela Merten [de], Marina Krauser, Louise Martini [de], Hannes Jaenicke, Klaus Schwarzkopf
List of German films of the 1990s
List_of_German_films_of_the_1990s
Wooden footpath to cross wet land
Rüdiger; Brandt, Jochen; Först, Elke; Krause, Yvonne; Merkel, Michael; Mertens, Kathrin; Weiss, Rainer-Maria (2013). Archaeological Museum Hamburg Helms-Museum:
Boardwalk
Solution concept of a non-cooperative game
building with great depth on such ideas Mertens-stable equilibria were introduced as a solution concept. Mertens stable equilibria satisfy both forward
Nash_equilibrium
MathWorld. Weisstein, Eric W. "Dottie Number". MathWorld. Weisstein, Eric W. "Mertens Constant". MathWorld. Weisstein, Eric W. "Universal Parabolic Constant"
List of mathematical constants
List_of_mathematical_constants
Dutch footballer (born 1994)
time for Zakaria Labyad. On 18 March, six minutes after replacing Dries Mertens, he scored his first league goal to confirm a 5–1 win over Heerenveen.
Memphis_Depay
1960–1963 unrecognised state in Africa
wealth, it was realized that Adoula government could not economically function if Katanga were allowed to secede, causing Kennedy to come down on the
State_of_Katanga
Mathematical models of strategic interactions
given probability distribution function. Therefore, the players maximize the mathematical expectation of the cost function. It was shown that the modified
Game_theory
Senegalese footballer (born 1991)
"FORMAZIONI UFFICIALI - Out Gargano, Kou e De Guzman, ci sono Britos, Inler e Mertens" (in Italian). Tutto Napoli. 16 April 2015. Archived from the original
Kalidou_Koulibaly
Dutch mathematician (born 1947)
non-trivial zeros of the Riemann zeta function with Jan van de Lune and Dik Winter, for disproving the Mertens conjecture with Andrew Odlyzko, and for
Herman_te_Riele
Dynamical system
dynamically on the distribution of population types, making the fitness function an endogenous component of the system. This allows it to model frequency-dependent
Replicator_equation
1/4} . Selberg's orthogonality conjecture: generalization of Mertens' theorem for functions in Selberg class. Bombieri–Lang conjecture: K {\displaystyle
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Decision rule used for minimizing the possible loss for a worst-case scenario
actions taken by all other players. v i {\displaystyle v_{i}} is the value function of player i. Calculating the maximin value of a player is done in a worst-case
Minimax
Palm tree with many uses, mainly fruit as cash crop
Web-Based Açai Scams". CSPI. Retrieved 2 September 2012. Pacheco-Palencia LA, Mertens-Talcott S, Talcott ST (June 2008). "Chemical composition, antioxidant properties
Açaí_palm
Species of comb jelly
the German naturalist Karl Heinrich Mertens aka Andrei Karlovich Mertens (17 May 1796 – 18 September 1830). Mertens accompanied the Russian naturalist
Mertensia_ovum
Diode that emits light from an organic compound
2012-01-25 at the Wayback Machine. Arstechnica.com. Retrieved 2011-10-04. Mertens, Ron; Tanalin, Marat (14 January 2018). "Pulse-width modulation (PWM) in
OLED
process, an (informal) institutional working practice, or an EU body, function or decision, and which is commonly understood among EU officials or external
Glossary of European Union concepts, acronyms, and jargon
Glossary_of_European_Union_concepts,_acronyms,_and_jargon
1798 (10): 1926–1933. doi:10.1016/j.bbamem.2010.06.029. PMID 20637181. Mertens, J; Aliyu, H; Cowan, DA (1 August 2018). "LEA Proteins and the Evolution
Late embryogenesis abundant proteins
Late_embryogenesis_abundant_proteins
Concept in game theory
strategy Nash equilibria. Reaction correspondences are not "reaction functions" since functions must only have one value per argument, and many reaction correspondences
Best_response
Intersection graph conjecture Kelvin's conjecture Kouchnirenko's conjecture Mertens conjecture Pólya conjecture, 1919 (1958) Ragsdale conjecture Schoenflies
List_of_conjectures
Base-1 numeral system
University Press, §17, pp. 32–33, retrieved May 10, 2017. Moore, Cristopher; Mertens, Stephan (2011), The Nature of Computation, Oxford University Press, p
Unary_numeral_system
Prime Minister of Hungary since 2026
as "not a punishment but a sign of social justice and solidarity in a functioning and humane country". His government has also planned to phase out the
Péter_Magyar
European Union committee
Council. In turn, COREPER's agenda is prepared by two bodies known as the Mertens Group and Antici Group, depending on the configuration. The COREPER is
Committee of Permanent Representatives
Committee_of_Permanent_Representatives
Permanent or temporary changes to human sex organs
MacDonald, Noni E.; McAllister, Ryan; Meddings, Jonathan; Merli, Claudia; Mertens, Mayli; Milos, Marilyn; Mishori, Ranit; Monro, Surya; Moss, Lisa Braver;
Genital modification and mutilation
Genital_modification_and_mutilation
Protein-coding gene in the species Homo sapiens
Caplan S, Mertens D, Hynes G, Pitluk Z, Kashi Y, Harrison-Lavoie K, Stevenson S, Brown C, Barrell B (1994). "Primary structure and function of a second
T-complex_1
Austrian mathematician (1875–1954)
was a mathematician born in Vienna, Austria. He worked alongside Franz Mertens and Hermann Minkowski at the universities of Vienna and Zurich, respectively
Ernst_Sigismund_Fischer
Species of venomous snake
in Thailand and is considered a synonym of D. siamensis. D. s. limitis (Mertens, 1927) occurs in Indonesia and is considered a synonym of D. siamensis
Russell's_viper
MERTENS FUNCTION
MERTENS FUNCTION
Boy/Male
Scandinavian
Warrior of Mars.
Girl/Female
Latin
Marvelous.
Male
Danish
, of Mars.
Boy/Male
African, American, Anglo, Australian, British, Christian, English
From the Town by the Lake
Surname or Lastname
Portuguese
Portuguese : patronymic from the personal name Martim, vernacular form of Latin Martinus (see Martin).English and Dutch : patronymic from the personal name Martin.
Male
Norwegian
Danish and Norwegian form of Latin Martinus, MORTEN means "of/like Mars."
Female
Egyptian
, Merte.
Boy/Male
Australian, Swedish
From the God Mars
Boy/Male
British, English
Small Falcon
Boy/Male
Anglo Saxon American English
From the farm by the sea.
Boy/Male
Australian, British, Danish, English, French, Swedish
From the Moor Town; From the God Mars
Girl/Female
British, English
Botanical Name; The Myrtle is a Dark Green Shrub with Pink or White Blossoms
Surname or Lastname
English
English : habitational name from places called Merton in London, Devon, Norfolk, and Oxfordshire, named in Old English with mere ‘lake’, ‘pool’ + tūn ‘enclosure’, ‘settlement’. Compare Marton, Martin 2.
Surname or Lastname
English
English : variant of Merrin.
Surname or Lastname
North German and Dutch
North German and Dutch : patronymic from Marten.English : variant of Martins.
Surname or Lastname
English (Devon)
English (Devon) : unexplained; perhaps a variant of Matters, itself a variant of Matter.
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : habitational name from Mortagne in La Manche, France. This surname may have been sometimes confused with Morton.
Boy/Male
Australian, British, Dutch, English, French, German, Latin, Netherlands, Scandinavian, Swedish
Warrior of Mars; Warlike; Little Marcus; Dedicated to Mars; Like Mars
Male
German
Low German form of French Martin, MERTEN means "of/like Mars."
Male
German
Low German form of Latin Martinus, MARTEN means "of/like Mars."
MERTENS FUNCTION
MERTENS FUNCTION
Male
English
Pet form of English Reuben, RUBY means "behold, a son." Compare with feminine Ruby.
Girl/Female
Indian
Agree, Comforter, Consoler
Boy/Male
Arabic, Muslim
Obedient; Submissive
Girl/Female
Hebrew
Grace.
Boy/Male
Indian, Sanskrit
Renowned; Famous
Surname or Lastname
English and Scottish
English and Scottish : patronymic from Pott 1, particularly common in northeastern England.
Girl/Female
Hindu, Indian, Traditional
Pleased
Boy/Male
Hindu, Indian, Traditional
Lord of the World
Boy/Male
Indian, Telugu
Lord Shiva
Boy/Male
Slavic
Stand of glory.
MERTENS FUNCTION
MERTENS FUNCTION
MERTENS FUNCTION
MERTENS FUNCTION
MERTENS FUNCTION
adv.
Certainly; in truth; verily.
n.
Any one of several fur-bearing carnivores of the genus Mustela, closely allied to the sable. Among the more important species are the European beech, or stone, marten (Mustela foina); the pine marten (M. martes); and the American marten, or sable (M. Americana), which some zoologists consider only a variety of the Russian sable.
n.
Same as Marten.
n.
Dactylic meters.
v. t.
To render fit.
pl.
of Germen
n.
A bird. See Martin.
n.
The fur of the marten, used for hats, muffs, etc.
n.
One who, or that which, heartens, animates, or stirs up.
pl.
of Merman
a.
Like or pertaining to the family Mustelidae, or the weasels and martens.
n.
A small South African antelope (Cephalous mergens); -- called also impoon, and deloo.
n.
The constellation Serpens.
n.
One who, or that which, merges.
a.
Covered with a mitten or mittens.
n.
A genus of birds including the swallows and martins.
a.
Having two poetical measures or meters.
n.
A constellation represented as a serpent held by Serpentarius.
n.
A verse of two meters.
n.
The beech marten (Mustela foina). See Marten.