In number theory, the Mertens function is defined for all positive integers n as where is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive real numbers as follows: Less formally, is the count of square-free integers up to x that have an even number of prime factors, minus the count of those that have an odd number. The first 143 M(n) values are (sequence in the OEIS) The Mertens function slowly grows in positive and negative directions both on average and in peak value, oscillating in an apparently chaotic manner passing through zero when n has the values 2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, 329, 331, 332, 333, 353, 355, 356, 358, 362, 363, 364, 366, 393, 401, 403, 404, 405, 407, 408, 413, 414, 419, 420, 422, 423, 424, 425, 427, 428, ... (sequence in the OEIS). Because the Möbius function only takes the values −1, 0, and +1, the Mertens function moves slowly, and there is no x such that |M(x)| > x. The Mertens conjecture went further, stating that there would be no x where the absolute value of the Mertens function exceeds the square root of x. The Mertens conjecture was proven false in 1985 by Andrew Odlyzko and Herman te Riele. However, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M(x), namely M(x) = O(x1/2 + ε). Since high values for M(x) grow at least as fast as , this puts a rather tight bound on its rate of growth. Here, O refers to big O notation. The true rate of growth of M(x) is not known. An unpublished conjecture of Steve Gonek states that Probabilistic evidence towards this conjecture is given by Nathan Ng. In particular, Ng gives a conditional proof that the function has a limiting distribution on . That is, for all bounded Lipschitz continuous functions on the reals we have that (Wikipedia).
The Weierstrass Definition of the GAMMA FUNCTION! - Proving Equivalence!
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From playlist Limits
Mertens Conjecture Disproof and the Riemann Hypothesis | MegaFavNumbers
#MegaFavNumbers The Mertens conjecture is a conjecture is a conjecture about the distribution of the prime numbers. It can be seen as a stronger version of the Riemann hypothesis. It says that the Mertens function is bounded by sqrt(n). The Riemann hypothesis on the other hand only require
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From playlist Acute Care Surgery
A Prime Surprise (Mertens Conjecture) - Numberphile
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From playlist Women in Mathematics - Numberphile
An Amazing Connection Between the Riemann Hypothesis and Topology
https://gregoriousmaths.com/2021/08/19/a-couple-of-other-connections-between-number-theory-and-topology/ 0:00 Introduction and plan 2:32 The Riemann hypothesis 7:22 Introducing the complex we will study 19:41 Studying the asymptotic behaviour of \beta_k(\Delta_n) 22:54 Some number theoret
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Risk Management Lesson 8A: Industrial Models for Credit Risk
In this first part of Lesson 8, we deal with two important credit risk models developed by the industry. Topics: - Moody's KMV - CreditMetrics (J.P. Morgan & Co.)
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From playlist Biomathematics
Inequalities with Polynomial Functions (Precalculus - College Algebra 46)
Support: https://www.patreon.com/ProfessorLeonard Professor Leonard Merch: https://professor-leonard.myshopify.com How to solve inequalities that involve polynomial functions beyond quadratic. Focus will be on solving inequalities with a graphical approach using multiplicity of x-intercep
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Definition, horizontal line test, and examples! Facebook: https://www.facebook.com/braingainzofficial Instagram: https://www.instagram.com/braingainzofficial Thanks for watching! Comment below with any questions / feedback and make sure to like / subscribe if you enjoyed!
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One to One Functions (Precalculus - College Algebra 50)
Support: https://www.patreon.com/ProfessorLeonard Professor Leonard Merch: https://professor-leonard.myshopify.com What one-to-one functions are, how to determine one-to-one-ness, and why they are important to inverse functions
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Composition of Functions (Precalculus - College Algebra 48)
Support: https://www.patreon.com/ProfessorLeonard Professor Leonard Merch: https://professor-leonard.myshopify.com A re-introduction to composition of functions and how to perform them.
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Euler-Mascheroni V: The Meissel-Mertens Constant
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From playlist Analysis
Liouville and JT Quantum Gravity - Holography and Matrix Models - Thomas Mertens
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Sigmoid functions for population growth and A.I.
Some elaborations on sigmoid functions. https://en.wikipedia.org/wiki/Sigmoid_function https://www.learnopencv.com/understanding-activation-functions-in-deep-learning/ If you have any questions of want to contribute to code or videos, feel free to write me a message on youtube or get my co
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The Selberg sieve (Lecture 1) by Stephan Baier
Program Workshop on Additive Combinatorics ORGANIZERS: S. D. Adhikari and D. S. Ramana DATE: 24 February 2020 to 06 March 2020 VENUE: Madhava Lecture Hall, ICTS Bangalore Additive combinatorics is an active branch of mathematics that interfaces with combinatorics, number theory, ergod
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What's so significant about the nudity in this early Greek sculpture? | Art, Explained
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From playlist Biomathematics
CTNT 2020 - Sieves (by Brandon Alberts) - Lecture 4
The Connecticut Summer School in Number Theory (CTNT) is a summer school in number theory for advanced undergraduate and beginning graduate students, to be followed by a research conference. For more information and resources please visit: https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2020 - Sieves (by Brandon Alberts)
Risk Management Lesson 7B: Credit Ratings (continued) and Merton's Model
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Introduction to Polynomial Functions (Precalculus - College Algebra 27)
Support: https://www.patreon.com/ProfessorLeonard Cool Mathy Merch: https://professor-leonard.myshopify.com What polynomial functions are, how they are organized, and their important features.
From playlist Precalculus - College Algebra/Trigonometry