Proof That The Sum Of The Reciprocals Of The Primes Diverges
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The sum of the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
s of all
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 diverges; that is: \sum_\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1 + \frac1 + \frac1 + \cdots = \infty This was proved by
Leonhard 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 ...
in 1737, and strengthens
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's 3rd-century-BC result that there are infinitely many prime numbers and
Nicole Oresme Nicole Oresme (; c. 1320–1325 – 11 July 1382), also known as Nicolas Oresme, Nicholas Oresme, or Nicolas d'Oresme, was a French philosopher of the later Middle Ages. He wrote influential works on economics, mathematics, physics, astrology an ...
's 14th-century proof of the divergence of the sum of the reciprocals of the integers (harmonic series). There are a variety of proofs of Euler's result, including a
lower bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element ...
for the partial sums stating that \sum_\frac1p \ge \log \log (n+1) - \log\frac6 for all natural numbers . The double
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
() indicates that the divergence might be very slow, which is indeed the case. See
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 ...
.


The harmonic series

First, we describe how Euler originally discovered the result. He was considering the harmonic series \sum_^\infty \frac = 1 + \frac + \frac + \frac + \cdots = \infty He had already used the following "
product formula In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuation (algebra), valuations. There are two kinds of global field (mathematics), fields: *Algebraic number field: A algebrai ...
" to show the existence of infinitely many primes. \sum_^\infty \frac = \prod_ \left( 1+\frac+\frac+\cdots \right) = \prod_ \frac Here the product is taken over the set of all primes. Such infinite products are today called
Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Eul ...
s. The product above is a reflection of the
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
. Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series.


Proofs


Euler's proof

Euler considered the above product formula and proceeded to make a sequence of audacious leaps of logic. First, he took the natural logarithm of each side, then he used the Taylor series expansion for as well as the sum of a converging series: \begin \log \left( \sum_^\infty \frac\right) & = \log\left( \prod_p \frac\right) = -\sum_p \log \left( 1-\frac\right) \\ pt & = \sum_p \left( \frac + \frac + \frac + \cdots \right) \\ pt & = \sum_\frac + \frac\sum_p \frac + \frac\sum_p \frac + \frac\sum_p \frac+ \cdots \\ pt & = A + \frac B+ \frac C+ \frac D + \cdots \\ pt & = A + K \end for a fixed constant . Then he invoked the relation \sum_^\infty\frac = \log\infty, which he explained, for instance in a later 1748 work, by setting in the Taylor series expansion \log\left(\frac1\right)=\sum_^\infty\fracn. This allowed him to conclude that A = \frac + \frac + \frac + \frac + \frac + \cdots = \log \log \infty. It is almost certain that Euler meant that the sum of the reciprocals of the primes less than is asymptotic to as approaches infinity. It turns out this is indeed the case, and a more precise version of this fact was rigorously 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 ...
in 1874. Thus Euler obtained a correct result by questionable means.


Erdős's proof by upper and lower estimates

The following
proof by contradiction In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as ...
comes from
Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
. Let denote the th prime number. Assume that the sum of the reciprocals of the primes converges. Then there exists a smallest
positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posit ...
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
such that \sum_^\infty \frac 1 < \frac12 \qquad(1) For a positive integer , let denote the set of those in which are not
divisible In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
by any prime greater than (or equivalently all which are a product of powers of primes ). We will now derive an upper and a lower estimate for , the number of elements in . For large , these bounds will turn out to be contradictory. ;Upper estimate: :Every in can be written as with positive integers and , where is
square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. A ...
. Since only the primes can show up (with exponent 1) in the
prime factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are suf ...
of , there are at most different possibilities for . Furthermore, there are at most possible values for . This gives us the upper estimate , M_x, \le 2^k\sqrt \qquad(2) ;Lower estimate: :The remaining numbers in the
set difference In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the ...
are all divisible by a prime greater than . Let denote the set of those in which are divisible by the th prime . Then \\setminus M_x = \bigcup_^\infty N_ :Since the number of integers in is at most (actually zero for ), we get x-, M_x, \le \sum_^\infty , N_, < \sum_^\infty \frac x :Using (1), this implies \frac x 2 < , M_x, \qquad(3) This produces a contradiction: when , the estimates (2) and (3) cannot both hold, because .


Proof that the series exhibits log-log growth

Here is another proof that actually gives a lower estimate for the partial sums; in particular, it shows that these sums grow at least as fast as . The proof is due to Ivan Niven, adapted from the product expansion idea of
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 ...
. In the following, a sum or product taken over always represents a sum or product taken over a specified set of primes. The proof rests upon the following four inequalities: * Every positive integer can be uniquely expressed as the product of a square-free integer and a square as a consequence of the
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
. Start with i = q_1^ \cdot q_2^ \cdots q_r^, where the ''β''s are 0 (the corresponding power of prime is even) or 1 (the corresponding power of prime is odd). Factor out one copy of all the primes whose β is 1, leaving a product of primes to even powers, itself a square. Relabeling: i = (p_1 p_2 \cdots p_s) \cdot b^2, where the first factor, a product of primes to the first power, is square free. Inverting all the s gives the inequality \sum_^n \frac 1 i \le \left(\prod_ \left(1 + \frac 1 p \right)\right) \cdot \left(\sum_^n \frac 1 \right) = A \cdot B. To see this, note that \frac 1 i = \frac 1 \cdot \frac 1 , and \begin \left(1 + \frac\right)\left(1 + \frac\right) \ldots \left(1 + \frac\right) &= \left(\frac\right)\left(\frac\right)\cdots\left(\frac\right) + \ldots\\ &= \frac 1 + \ldots. \end That is, 1/(p_1p_2 \cdots p_s) is one of the summands in the expanded product . And since 1 / b^2 is one of the summands of , every summand 1/i is represented in one of the terms of when multiplied out. The inequality follows. * The upper estimate for the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
\begin \log(n+1) &= \int_1^ \fracx \\ &= \sum_^n\underbrace_ - \frac1\right)}_ \\ &= 1 + \frac23 - \frac1 < \frac53 \end Combining all these inequalities, we see that \begin \log(n+1) & < \sum_^n\frac \\ & \le \prod_ \left(1 + \frac\right) \sum_^n \frac \\ & < \frac53\prod_ \exp\left(\frac\right) \\ & = \frac53\exp\left(\sum_ \frac \right) \end Dividing through by and taking the natural logarithm of both sides gives \log\log(n + 1) - \log\frac53 < \sum_ \frac as desired.  Q.E.D. Using \sum_^\infty \frac = \frac6 (see the
Basel problem The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
), the above constant can be improved to ; in fact it turns out that \lim_ \left( \sum_ \frac - \log \log n \right) = M where 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 ...
(somewhat analogous to the much more famous
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 ...
).


Proof from Dusart's inequality

From Dusart's inequality, we get p_n < n \log n + n \log \log n \quad\mbox n \ge 6 Then \begin \sum_^\infty \frac1 &\ge \sum_^\infty \frac \\ &\ge \sum_^\infty \frac \\ &\ge \sum_^\infty \frac = \infty \end by the
integral test for convergence In mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test. ...
. This shows that the series on the left diverges.


Geometric and harmonic-series proof

Suppose for contradiction the sum converged. Then, there exists n such that \sum_ \frac < 1 . Call this sum x . Now consider the convergent geometric series x + x^2 + x^3 + \cdots. This geometric series contains the sum of reciprocals of all numbers whose prime factorization contain only primes in the set \. Consider the subseries \sum_ \frac. This is a subseries because 1 + i(p_1p_2 \cdots p_n) is not divisible by any p_j, j \leq n . However, by the
Limit comparison test In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series. Statement Suppose that we have two series \Sigma_n a_n and \Sigma_n b_n ...
, this subseries diverges by comparing it to the harmonic series. Indeed, \lim_ \frac = p_1 p_2 \cdots p_n. Thus, we have found a divergent subseries of the original convergent series, and since all terms are positive, this gives the contradiction. We may conclude \sum_ \frac diverges. Q.E.D.


Partial sums

While the
partial sum In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
s of the reciprocals of the primes eventually exceed any integer value, they never equal an integer. One proof is by induction: The first partial sum is , which has the form . If the th partial sum (for ) has the form , then the st sum is \frac\text\text + \frac = \frac = \frac\text = \frac\text\text as the st prime is odd; since this sum also has an form, this partial sum cannot be an integer (because 2 divides the denominator but not the numerator), and the induction continues. Another proof rewrites the expression for the sum of the first reciprocals of primes (or indeed the sum of the reciprocals of ''any'' set of primes) in terms of the
least common denominator In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions. Description The l ...
, which is the product of all these primes. Then each of these primes divides all but one of the numerator terms and hence does not divide the numerator itself; but each prime ''does'' divide the denominator. Thus the expression is irreducible and is non-integer.


See also

*
Euclid's theorem Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work '' Elements''. There are several proofs of the theorem. Euclid's proof Euclid offere ...
that there are infinitely many primes *
Small set (combinatorics) In combinatorial mathematics, a large set of positive integers :S = \ is one such that the infinite sum of the reciprocals :\frac+\frac+\frac+\frac+\cdots diverges. A small set is any subset of the positive integers that is not large; that i ...
*
Brun's theorem In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by ''B''2 . Brun's theorem was proved by V ...
, on the convergent sum of reciprocals of the twin primes *
List of sums of reciprocals In mathematics and especially number theory, the sum of reciprocals generally is computed for the reciprocals of some or all of the positive integers (counting numbers)—that is, it is generally the sum of unit fractions. If infinitely many n ...


References

Sources *


External links

* {{DEFAULTSORT:Sum Of The Reciprocals Of The Primes Diverges Mathematical series Articles containing proofs Theorems about prime numbers