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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
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
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
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)
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. It is the sum of five consecutive primes (59 + 61 + 67 + 71 + 73). 332 = 22 × 83. It is a zero of Mertens function and number
300_(number)
Natural number
number, a zero of Mertens function, and the smallest number that is the product of an emirp pair. 404 = 22 × 101, a zero of Mertens function, a nontotient
400_(number)
Natural number
sum of the first three powers of 3 (31 + 32 + 33). 39 is a zero of Mertens function. 39 has an aliquot sum of 17, which is a prime. 39 is the 4th member
39_(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)
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
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
centered triangular number and a zero of Mertens function. 1220 = 22 × 5 × 61. It is a zero of Mertens function. There are 1220 binary vectors of length
1000_(number)
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
Mertens function, and the 42nd triangular number. 904 = 23 × 113. It is a refactorable number, a lazy caterer number, and a zero of Mertens function.
900_(number)
Natural number
(111001112). There are 231 integer partitions of 16. 231 is a zero of Mertens function. A US gallon is defined by being exactly 231 cubic inches, the volume
231_(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)
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)
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
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
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
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)
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
Natural number
natural number following 419 and preceding 421. 420 is a zero of the Mertens function, a sparsely totient number, a pronic number, a balanced number, a largely
420_(number)
Natural number
composite Thabit number. 95 is the smallest integer for which the Mertens function is greater than 1. In statistics, a 95% confidence interval is considered
95_(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
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)
Mathematical constant
In mathematics, the Meissel–Mertens constant, named after Ernst Meissel and Franz Mertens, also referred to as the Mertens constant, Kronecker's constant
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
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
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
[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)
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
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
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)
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
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)
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
Hilbert–Pólya conjecture Generalized Riemann hypothesis Mertens function, Mertens conjecture, Meissel–Mertens constant De Bruijn–Newman constant Dirichlet character
List_of_number_theory_topics
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
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
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
Natural number
2301 – nonagonal number 2309 – primorial prime, twin prime with 2311, Mertens function zero, highly cototient number 2310 – fifth primorial 2311 – primorial
2000_(number)
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
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
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
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
Belgian-French writer and lawyer (1939–2025)
Chevalier de l'Ordre des Arts et des Lettres by France. Mertens reflected much on the social function of the writer. For him, private life, fiction, and history
Pierre_Mertens
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
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
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
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
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
Disease where the kidneys fail to adequately filter waste products from the blood
McGraw Hill Medical. p. 733. ISBN 978-0-07-145153-6. Scurt FG, Bose K, Mertens PR, Chatzikyrkou C, Herzog C (2024-06-01). "Cardiac Surgery-Associated
Kidney_failure
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
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
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
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
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
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
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
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
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
Dutch optician
early microscope which he also attributed to Janssen, perpetuating the Janssen claim to both devices. Its actual function and creator has been disputed.
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
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
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
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
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
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
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
Mathematical concept in number theory
for the Riemann Zeta function, several advancements relating to Partial Euler Products have been made. It mainly began with Mertens' 3rd Theorem (as seen
Partial_Euler_Product
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
Medical condition
Moreover, these antibodies have been demonstrated to reduce potassium channel function in neuronal cell lines. It is suspected that the peripheral nerve hyperexcitability
Neuromyotonia
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
Photographic and cinematographic technique using a large depth of field
Focus Cinematography in the United States | Screen | Oxford Academic". Mertens, Jacob (28 November 2013). "3:10 to Yuma (1957)". Film International: Thinking
Deep_focus
Membrane that surrounds or partially covers the vaginal opening
Wilkins. ISBN 978-0-7817-8249-4. Lardenoije, Céline; Aardenburg, Robert; Mertens, Helen (May 26, 2009). "Imperforate hymen: a cause of abdominal pain in
Hymen
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
Mathematical models of strategic interactions
alternatives x , y ∈ X {\displaystyle x,y\in X} , a utility function on X {\displaystyle X} is a function u → R {\displaystyle u\to \mathbb {R} } associating
Game_theory
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
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
Genus of kelp
German-Russian explorer Fyodor Petrovich Litke (also spelled Lütke) by Mertens. The species was renamed in a description by Postels and Ruprecht. Nereocystis
Nereocystis
German state election
was held on the same day, as well as the Borough council elections who function as the municipal elections of Berlin. The SPD remained the largest party
2021_Berlin_state_election
10. Pandolfini 2009, p. 301. Seirawan & Silman 1994, p. 241. Moore & Mertens 2011, p. 14. Alburt & Parr 2003, pp. 22–23. Brace 1977. New Oxford American
Glossary_of_chess
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
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
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
Government gazette of the German Empire (1871–1945)
Hess 1998, pp. 34–35. Lukas 1903, p. 144. Jastrow 1925, p. 42. Mertens 2009, p. 62. Mertens 2009, p. 72. Jastrow 1925, pp. 40–41. Schneider 2002, marginal
Reichsgesetzblatt
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
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
Combinatorial game theory theorem
value, or Grundy number, and the function that assigns this value to each such position is called the Sprague–Grundy function. R. L. Sprague and P. M. Grundy
Sprague–Grundy_theorem
Chemical element with atomic number 74 (W)
Pamela, J.; Altmann, H.; Coad, J.P.; Hirai, T.; Lioure, A.; Maier, H.; Mertens, Ph.; Philipps, V.; Riccardo, V.; Rubel, M.; Villedieu, E.; JET ITER-like
Tungsten
Representation of a game in game theory
game, regardless of whether that stage actually arises in play. A payoff function for a player is a mapping from the cross-product of players' strategy spaces
Normal-form_game
Hungarian and American mathematician and physicist (1903–1957)
subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the
John_von_Neumann
German mathematician (1810–1893)
Ernst Eduard (1975), Weil, André (ed.), Collected papers. Volume II: Function theory, geometry and miscellaneous, Berlin, New York: Springer-Verlag,
Ernst_Kummer
Dutch footballer and musician (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
travel, tourism, insurance
MERTENS FUNCTION
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travel, tourism, insurance