In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, Mertens' theorems are three 1874 results related to the density of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s proved by
Franz Mertens
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 Posen, Kingdom of Prussia (now Środa Wielkopolska, Poland) and died in Vienna, Austria.
Th ...
.
[F. Mertens. J. reine angew. Math. 78 (1874), 46–6]
Ein Beitrag zur analytischen Zahlentheorie
/ref> "Mertens' theorem" may also refer to his theorem in analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
.
Theorems
In the following, let mean all primes not exceeding ''n''.
Mertens' first theorem:
:
does not exceed 2 in absolute value for any . ()
Mertens' second theorem:
:
where ''M'' is the Meissel–Mertens constant
The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as Mertens constant, Kronecker's constant, Hadamard– de la Vallée-Poussin constant or the prime reciprocal constant, is a mathematical constant in n ...
(). More precisely, Mertens proves that the expression under the limit does not in absolute value exceed
:
for any .
Mertens' third theorem:
:
where γ is the Euler–Mascheroni constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural l ...
().
Changes in sign
In a paper on the growth rate of the sum-of-divisors function
In mathematics, and specifically in number 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 of divisors of an integer'' (including ...
published in 1983, Guy Robin
Guy or GUY may refer to:
Personal names
* Guy (given name)
* Guy (surname)
* That Guy (...), the New Zealand street performer Leigh Hart
Places
* Guy, Alberta, a Canadian hamlet
* Guy, Arkansas, US, a city
* Guy, Indiana, US, an unincorpo ...
proved that in Mertens' 2nd theorem the difference
:
changes sign infinitely often, and that in Mertens' 3rd theorem the difference
:
changes sign infinitely often. Robin's results are analogous to Littlewood's famous theorem that the difference π(''x'') − li(''x'') changes sign infinitely often. No analog of the Skewes number
In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which
:\pi(x) > \operatorname(x),
where is the prime-counting function ...
(an upper bound on the first natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
''x'' for which π(''x'') > li(''x'')) is known in the case of Mertens' 2nd and 3rd theorems.
Mertens' second theorem and the prime number theorem
Regarding this asymptotic formula Mertens refers in his paper to "two curious formula of Legendre", the first one being Mertens' second theorem's prototype (and the second one being Mertens' third theorem's prototype: see the very first lines of the paper). He recalls that it is contained in Legendre's third edition of his "Théorie des nombres" (1830; it is in fact already mentioned in the second edition, 1808), and also that a more elaborate version was proved by Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebyshe ...
in 1851. Note that, already in 1737, Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
knew the asymptotic behaviour of this sum.
Mertens diplomatically describes his proof as more precise and rigorous. In reality none of the previous proofs are acceptable by modern standards: Euler's computations involve the infinity (and the hyperbolic logarithm of infinity, and the logarithm of the logarithm of infinity!); Legendre's argument is heuristic; and Chebyshev's proof, although perfectly sound, makes use of the Legendre-Gauss conjecture, which was not proved until 1896 and became better known as the prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
.
Mertens' proof does not appeal to any unproved hypothesis (in 1874), and only to elementary real analysis. It comes 22 years before the first proof of the prime number theorem which, by contrast, relies on a careful analysis of the behavior of the Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
as a function of a complex variable.
Mertens' proof is in that respect remarkable. Indeed, with modern notation it yields
:
whereas the prime number theorem (in its simplest form, without error estimate), can be shown to be equivalent to
:
In 1909 Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis.
Biography
Edmund Landau was born to a Jewish family in Berlin. His father was Leopold ...
, by using the best version of the prime number theorem then at his disposition, proved[Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig 1909, Repr. Chelsea New York 1953, § 55, p. 197-203.] that
:
holds; in particular the error term is smaller than for any fixed integer ''k''. A simple summation by parts
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformati ...
exploiting the strongest form known of the prime number theorem improves this to
:
for some .
Similarly a partial summation shows that is equivalent to the PNT.
Mertens' third theorem and sieve theory
An estimate of the probability of () having no factor is given by
:
This is closely related to Mertens' third theorem which gives an asymptotic approximation of
:
Proof
The main step in the proof of Mertens' second theorem is
:
where the last equality needs which follows from .
Thus, we have proved that
:.
A partial summation
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformati ...
yields
:.
References
Further reading
* Yaglom and Yaglom ''Challenging mathematical problems with elementary solutions'' Vol 2, problems 171, 173, 174
External links
*
*
*{{MathWorld, urlname=MertensSecondTheorem, title=Mertens' Second Theorem
*Varun Rajkumar
π(x) and the Sieve of Eratosthenes
Mathematical series
Summability theory
Theorems about prime numbers