analytic number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens. In the following, let...

For Mertens' results on the distribution of prime numbers, see Mertens' theorems. For Mertens' result on convergence of Cauchy products of series, see...

Franz Mertens (20 March 1840 – 5 March 1927) (also known as Franciszek Mertens) was a Polish mathematician. He was born in Schroda in the Grand Duchy of...

(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}}...

In mathematics, the Mertens conjecture is the statement that the Mertens function M ( n ) {\displaystyle M(n)} is bounded by ± n {\displaystyle \pm {\sqrt...

theorems, and Meissel–Mertens constant Franz Carl Mertens (1764–1831), German botanist Gregory Mertens (1991–2015), Belgian footballer Helmut Mertens...

called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as theorems only the...

This is a list of notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures...

quantum field theory The calculation of the Meissel–Mertens constant The third of Mertens' theorems* Solution of the second kind to Bessel's equation In...

The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as Mertens constant, Kronecker's constant, Hadamard–de la...

confirmed by later mathematicians as one of Mertens' theorems, and can be seen as a precursor to the prime number theorem. Another problem in number theory closely...

^{h}x}}\right)\ .} The Mertens conjecture went further, stating that there would be no x where the absolute value of the Mertens function exceeds the square...

of the first known proofs of theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle...

extrapolation Series acceleration Steffensen's method Hugh J. Hamilton, "Mertens' Theorem and Sequence Transformations", AMS (1947) Transformations of Integer...

two copies of this power series in the Cauchy sense, permitted by Mertens' theorem, shows that the defining multiplicative property of exponential functions...

9, pp. 160–188, Theorems 7 and 8. In Theorem 7 Euler proves the formula in the special case s = 1 {\displaystyle s=1} , and in Theorem 8 he proves it more...

(for example, the Paris–Harrington theorem) provable using second order but not first-order methods, but such theorems are rare to date. Erdős and Selberg's...

The 100th prime number is 541, which returns 0 {\displaystyle 0} for the Mertens function. It is the 10th star number (whose digit sum also adds to 10 in...

limitations": ...the magnitudes involved should lead one to suspect that theorems and arguments based chiefly on the mere finiteness [of] the state diagram...

while the second was published in 2020. Similar identities hold for the Mertens function. The formula ∑ d ∣ n μ ( d ) = { 1 if  n = 1 , 0 if  n > 1 {\displaystyle...

member, and getting the previous value before a member. Using one of Mertens' theorems (the third) it can be shown to use O ( N / log ⁡ log ⁡ N ) {\displaystyle...

hypothesis Critical line theorem Hilbert–Pólya conjecture Generalized Riemann hypothesis Mertens function, Mertens conjecture, Meissel–Mertens constant De Bruijn–Newman...

building with great depth on such ideas Mertens-stable equilibria were introduced as a solution concept. Mertens stable equilibria satisfy both forward...

ISBN 978-0-486-81690-6. For the Sylow theorems see p. 43; for Lagrange's theorem, see p. 12; for Burnside's theorem see p. 143. Bryant, John; Sangwin, Christopher...

Jean-François Mertens (11 March 1946 – 17 July 2012) was a Belgian game theorist and mathematical economist. Mertens contributed to economic theory in...

the United States. Franciszek Mertens, mathematician known for Mertens function, Mertens conjecture, Mertens's theorems. Josef Hofmann, designer of first...

the supervisor of Kurt Hensel, Adolf Kneser, Mathias Lerch, and Franz Mertens, amongst others. His philosophical view of mathematics put him in conflict...

where 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...

Intel. Divergence of the sum of the reciprocals of the primes Meissel–Mertens constant Nicely, Thomas R. (18 January 2010). "Enumeration to 1.6*10^15...

as of September 2022[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic...