A math function by Srinivasa Ramanujan
In number theory , Ramanujan's sum , usually denoted cq (n ), is a function of two positive integer variables q and n defined by the formula
c q ( n ) = ∑ 1 ≤ a ≤ q ( a , q ) = 1 e 2 π i a q n , {\displaystyle c_{q}(n)=\sum _{1\leq a\leq q \atop (a,q)=1}e^{2\pi i{\tfrac {a}{q}}n},} where (a , q ) = 1 means that a only takes on values coprime to q .
Srinivasa Ramanujan mentioned the sums in a 1918 paper.[ 1] In addition to the expansions discussed in this article, Ramanujan's sums are used in the proof of Vinogradov's theorem that every sufficiently large odd number is the sum of three primes .[ 2]
For integers a and b , a ∣ b {\displaystyle a\mid b} is read "a divides b " and means that there is an integer c such that b a = c . {\displaystyle {\frac {b}{a}}=c.} Similarly, a ∤ b {\displaystyle a\nmid b} is read "a does not divide b ". The summation symbol
∑ d ∣ m f ( d ) {\displaystyle \sum _{d\,\mid \,m}f(d)} means that d goes through all the positive divisors of m , e.g.
∑ d ∣ 12 f ( d ) = f ( 1 ) + f ( 2 ) + f ( 3 ) + f ( 4 ) + f ( 6 ) + f ( 12 ) . {\displaystyle \sum _{d\,\mid \,12}f(d)=f(1)+f(2)+f(3)+f(4)+f(6)+f(12).} ( a , b ) {\displaystyle (a,\,b)} is the greatest common divisor ,
ϕ ( n ) {\displaystyle \phi (n)} is Euler's totient function ,
μ ( n ) {\displaystyle \mu (n)} is the Möbius function , and
ζ ( s ) {\displaystyle \zeta (s)} is the Riemann zeta function .
These formulas come from the definition, Euler's formula e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} and elementary trigonometric identities.
c 1 ( n ) = 1 c 2 ( n ) = cos n π c 3 ( n ) = 2 cos 2 3 n π c 4 ( n ) = 2 cos 1 2 n π c 5 ( n ) = 2 cos 2 5 n π + 2 cos 4 5 n π c 6 ( n ) = 2 cos 1 3 n π c 7 ( n ) = 2 cos 2 7 n π + 2 cos 4 7 n π + 2 cos 6 7 n π c 8 ( n ) = 2 cos 1 4 n π + 2 cos 3 4 n π c 9 ( n ) = 2 cos 2 9 n π + 2 cos 4 9 n π + 2 cos 8 9 n π c 10 ( n ) = 2 cos 1 5 n π + 2 cos 3 5 n π {\displaystyle {\begin{aligned}c_{1}(n)&=1\\c_{2}(n)&=\cos n\pi \\c_{3}(n)&=2\cos {\tfrac {2}{3}}n\pi \\c_{4}(n)&=2\cos {\tfrac {1}{2}}n\pi \\c_{5}(n)&=2\cos {\tfrac {2}{5}}n\pi +2\cos {\tfrac {4}{5}}n\pi \\c_{6}(n)&=2\cos {\tfrac {1}{3}}n\pi \\c_{7}(n)&=2\cos {\tfrac {2}{7}}n\pi +2\cos {\tfrac {4}{7}}n\pi +2\cos {\tfrac {6}{7}}n\pi \\c_{8}(n)&=2\cos {\tfrac {1}{4}}n\pi +2\cos {\tfrac {3}{4}}n\pi \\c_{9}(n)&=2\cos {\tfrac {2}{9}}n\pi +2\cos {\tfrac {4}{9}}n\pi +2\cos {\tfrac {8}{9}}n\pi \\c_{10}(n)&=2\cos {\tfrac {1}{5}}n\pi +2\cos {\tfrac {3}{5}}n\pi \\\end{aligned}}} and so on (OEIS : A000012 , OEIS : A033999 , OEIS : A099837 , OEIS : A176742 ,.., OEIS : A100051 ,...). cq (n ) is always an integer.
Let ζ q = e 2 π i q . {\displaystyle \zeta _{q}=e^{\frac {2\pi i}{q}}.} Then ζq is a root of the equation xq − 1 = 0 . Each of its powers,
ζ q , ζ q 2 , … , ζ q q − 1 , ζ q q = ζ q 0 = 1 {\displaystyle \zeta _{q},\zeta _{q}^{2},\ldots ,\zeta _{q}^{q-1},\zeta _{q}^{q}=\zeta _{q}^{0}=1} is also a root. Therefore, since there are q of them, they are all of the roots. The numbers ζ q n {\displaystyle \zeta _{q}^{n}} where 1 ≤ n ≤ q are called the q -th roots of unity . ζq is called a primitive q -th root of unity because the smallest value of n that makes ζ q n = 1 {\displaystyle \zeta _{q}^{n}=1} is q . The other primitive q -th roots of unity are the numbers ζ q a {\displaystyle \zeta _{q}^{a}} where (a , q ) = 1. Therefore, there are φ(q ) primitive q -th roots of unity.
Thus, the Ramanujan sum cq (n ) is the sum of the n -th powers of the primitive q -th roots of unity.
It is a fact[ 3] that the powers of ζq are precisely the primitive roots for all the divisors of q .
Example. Let q = 12. Then
ζ 12 , ζ 12 5 , ζ 12 7 , {\displaystyle \zeta _{12},\zeta _{12}^{5},\zeta _{12}^{7},} and ζ 12 11 {\displaystyle \zeta _{12}^{11}} are the primitive twelfth roots of unity, ζ 12 2 {\displaystyle \zeta _{12}^{2}} and ζ 12 10 {\displaystyle \zeta _{12}^{10}} are the primitive sixth roots of unity, ζ 12 3 = i {\displaystyle \zeta _{12}^{3}=i} and ζ 12 9 = − i {\displaystyle \zeta _{12}^{9}=-i} are the primitive fourth roots of unity, ζ 12 4 {\displaystyle \zeta _{12}^{4}} and ζ 12 8 {\displaystyle \zeta _{12}^{8}} are the primitive third roots of unity, ζ 12 6 = − 1 {\displaystyle \zeta _{12}^{6}=-1} is the primitive second root of unity, and ζ 12 12 = 1 {\displaystyle \zeta _{12}^{12}=1} is the primitive first root of unity. Therefore, if
η q ( n ) = ∑ k = 1 q ζ q k n {\displaystyle \eta _{q}(n)=\sum _{k=1}^{q}\zeta _{q}^{kn}} is the sum of the n -th powers of all the roots, primitive and imprimitive,
η q ( n ) = ∑ d ∣ q c d ( n ) , {\displaystyle \eta _{q}(n)=\sum _{d\mid q}c_{d}(n),} and by Möbius inversion ,
c q ( n ) = ∑ d ∣ q μ ( q d ) η d ( n ) . {\displaystyle c_{q}(n)=\sum _{d\mid q}\mu \left({\frac {q}{d}}\right)\eta _{d}(n).} It follows from the identity x q − 1 = (x − 1)(x q −1 + x q −2 + ... + x + 1) that
η q ( n ) = { 0 q ∤ n q q ∣ n {\displaystyle \eta _{q}(n)={\begin{cases}0&q\nmid n\\q&q\mid n\\\end{cases}}} and this leads to the formula
c q ( n ) = ∑ d ∣ ( q , n ) μ ( q d ) d , {\displaystyle c_{q}(n)=\sum _{d\mid (q,n)}\mu \left({\frac {q}{d}}\right)d,} published by Kluyver in 1906.[ 4]
This shows that c q (n ) is always an integer. Compare it with the formula
ϕ ( q ) = ∑ d ∣ q μ ( q d ) d . {\displaystyle \phi (q)=\sum _{d\mid q}\mu \left({\frac {q}{d}}\right)d.} It is easily shown from the definition that c q (n ) is multiplicative when considered as a function of q for a fixed value of n :[ 5] i.e.
If ( q , r ) = 1 then c q ( n ) c r ( n ) = c q r ( n ) . {\displaystyle {\mbox{If }}\;(q,r)=1\;{\mbox{ then }}\;c_{q}(n)c_{r}(n)=c_{qr}(n).} From the definition (or Kluyver's formula) it is straightforward to prove that, if p is a prime number,
c p ( n ) = { − 1 if p ∤ n ϕ ( p ) if p ∣ n , {\displaystyle c_{p}(n)={\begin{cases}-1&{\mbox{ if }}p\nmid n\\\phi (p)&{\mbox{ if }}p\mid n\\\end{cases}},} and if p k is a prime power where k > 1,
c p k ( n ) = { 0 if p k − 1 ∤ n − p k − 1 if p k − 1 ∣ n and p k ∤ n ϕ ( p k ) if p k ∣ n . {\displaystyle c_{p^{k}}(n)={\begin{cases}0&{\mbox{ if }}p^{k-1}\nmid n\\-p^{k-1}&{\mbox{ if }}p^{k-1}\mid n{\mbox{ and }}p^{k}\nmid n\\\phi (p^{k})&{\mbox{ if }}p^{k}\mid n\\\end{cases}}.} This result and the multiplicative property can be used to prove
c q ( n ) = μ ( q ( q , n ) ) ϕ ( q ) ϕ ( q ( q , n ) ) . {\displaystyle c_{q}(n)=\mu \left({\frac {q}{(q,n)}}\right){\frac {\phi (q)}{\phi \left({\frac {q}{(q,n)}}\right)}}.} This is called von Sterneck's arithmetic function.[ 6] The equivalence of it and Ramanujan's sum is due to Hölder.[ 7] [ 8]
Other properties of c q (n )[ edit ] For all positive integers q ,
c 1 ( q ) = 1 c q ( 1 ) = μ ( q ) c q ( q ) = ϕ ( q ) c q ( m ) = c q ( n ) for m ≡ n ( mod q ) {\displaystyle {\begin{aligned}c_{1}(q)&=1\\c_{q}(1)&=\mu (q)\\c_{q}(q)&=\phi (q)\\c_{q}(m)&=c_{q}(n)&&{\text{for }}m\equiv n{\pmod {q}}\\\end{aligned}}} For a fixed value of q the absolute value of the sequence { c q ( 1 ) , c q ( 2 ) , … } {\displaystyle \{c_{q}(1),c_{q}(2),\ldots \}} is bounded by φ(q ), and for a fixed value of n the absolute value of the sequence { c 1 ( n ) , c 2 ( n ) , … } {\displaystyle \{c_{1}(n),c_{2}(n),\ldots \}} is bounded by n .
If q > 1
∑ n = a a + q − 1 c q ( n ) = 0. {\displaystyle \sum _{n=a}^{a+q-1}c_{q}(n)=0.} Let m 1 , m 2 > 0, m = lcm(m 1 , m 2 ). Then[ 9] Ramanujan's sums satisfy an orthogonality property :
1 m ∑ k = 1 m c m 1 ( k ) c m 2 ( k ) = { ϕ ( m ) m 1 = m 2 = m , 0 otherwise {\displaystyle {\frac {1}{m}}\sum _{k=1}^{m}c_{m_{1}}(k)c_{m_{2}}(k)={\begin{cases}\phi (m)&m_{1}=m_{2}=m,\\0&{\text{otherwise}}\end{cases}}} Let n , k > 0. Then[ 10]
∑ gcd ( d , k ) = 1 d ∣ n d μ ( n d ) ϕ ( d ) = μ ( n ) c n ( k ) ϕ ( n ) , {\displaystyle \sum _{\stackrel {d\mid n}{\gcd(d,k)=1}}d\;{\frac {\mu ({\tfrac {n}{d}})}{\phi (d)}}={\frac {\mu (n)c_{n}(k)}{\phi (n)}},} known as the Brauer - Rademacher identity.
If n > 0 and a is any integer, we also have[ 11]
∑ gcd ( k , n ) = 1 1 ≤ k ≤ n c n ( k − a ) = μ ( n ) c n ( a ) , {\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}c_{n}(k-a)=\mu (n)c_{n}(a),} due to Cohen.
Ramanujan sum c s (n ) n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 s 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 3 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 4 0 −2 0 2 0 −2 0 2 0 −2 0 2 0 −2 0 2 0 −2 0 2 0 −2 0 2 0 −2 0 2 0 −2 5 −1 −1 −1 −1 4 −1 −1 −1 −1 4 −1 −1 −1 −1 4 −1 −1 −1 −1 4 −1 −1 −1 −1 4 −1 −1 −1 −1 4 6 1 −1 −2 −1 1 2 1 −1 −2 −1 1 2 1 −1 −2 −1 1 2 1 −1 −2 −1 1 2 1 −1 −2 −1 1 2 7 −1 −1 −1 −1 −1 −1 6 −1 −1 −1 −1 −1 −1 6 −1 −1 −1 −1 −1 −1 6 −1 −1 −1 −1 −1 −1 6 −1 −1 8 0 0 0 −4 0 0 0 4 0 0 0 −4 0 0 0 4 0 0 0 −4 0 0 0 4 0 0 0 −4 0 0 9 0 0 −3 0 0 −3 0 0 6 0 0 −3 0 0 −3 0 0 6 0 0 −3 0 0 −3 0 0 6 0 0 −3 10 1 −1 1 −1 −4 −1 1 −1 1 4 1 −1 1 −1 −4 −1 1 −1 1 4 1 −1 1 −1 −4 −1 1 −1 1 4 11 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 10 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 10 −1 −1 −1 −1 −1 −1 −1 −1 12 0 2 0 −2 0 −4 0 −2 0 2 0 4 0 2 0 −2 0 −4 0 −2 0 2 0 4 0 2 0 −2 0 −4 13 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 12 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 12 −1 −1 −1 −1 14 1 −1 1 −1 1 −1 −6 −1 1 −1 1 −1 1 6 1 −1 1 −1 1 −1 −6 −1 1 −1 1 −1 1 6 1 −1 15 1 1 −2 1 −4 −2 1 1 −2 −4 1 −2 1 1 8 1 1 −2 1 −4 −2 1 1 −2 −4 1 −2 1 1 8 16 0 0 0 0 0 0 0 −8 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 −8 0 0 0 0 0 0 17 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 16 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 18 0 0 3 0 0 −3 0 0 −6 0 0 −3 0 0 3 0 0 6 0 0 3 0 0 −3 0 0 −6 0 0 −3 19 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 18 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 20 0 2 0 −2 0 2 0 −2 0 −8 0 −2 0 2 0 −2 0 2 0 8 0 2 0 −2 0 2 0 −2 0 −8 21 1 1 −2 1 1 −2 −6 1 −2 1 1 −2 1 −6 −2 1 1 −2 1 1 12 1 1 −2 1 1 −2 −6 1 −2 22 1 −1 1 −1 1 −1 1 −1 1 −1 −10 −1 1 −1 1 −1 1 −1 1 −1 1 10 1 −1 1 −1 1 −1 1 −1 23 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 22 −1 −1 −1 −1 −1 −1 −1 24 0 0 0 4 0 0 0 −4 0 0 0 −8 0 0 0 −4 0 0 0 4 0 0 0 8 0 0 0 4 0 0 25 0 0 0 0 −5 0 0 0 0 −5 0 0 0 0 −5 0 0 0 0 −5 0 0 0 0 20 0 0 0 0 −5 26 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 −12 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 12 1 −1 1 −1 27 0 0 0 0 0 0 0 0 −9 0 0 0 0 0 0 0 0 −9 0 0 0 0 0 0 0 0 18 0 0 0 28 0 2 0 −2 0 2 0 −2 0 2 0 −2 0 −12 0 −2 0 2 0 −2 0 2 0 −2 0 2 0 12 0 2 29 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 28 −1 30 −1 1 2 1 4 −2 −1 1 2 −4 −1 −2 −1 1 −8 1 −1 −2 −1 −4 2 1 −1 −2 4 1 2 1 −1 8
Ramanujan expansions [ edit ] If f (n ) is an arithmetic function (i.e. a complex-valued function of the integers or natural numbers), then a convergent infinite series of the form:
f ( n ) = ∑ q = 1 ∞ a q c q ( n ) {\displaystyle f(n)=\sum _{q=1}^{\infty }a_{q}c_{q}(n)} or of the form:
f ( q ) = ∑ n = 1 ∞ a n c q ( n ) {\displaystyle f(q)=\sum _{n=1}^{\infty }a_{n}c_{q}(n)} where the ak ∈ C , is called a Ramanujan expansion [ 12] of f (n ).
Ramanujan found expansions of some of the well-known functions of number theory. All of these results are proved in an "elementary" manner (i.e. only using formal manipulations of series and the simplest results about convergence).[ 13] [ 14] [ 15]
The expansion of the zero function depends on a result from the analytic theory of prime numbers, namely that the series
∑ n = 1 ∞ μ ( n ) n {\displaystyle \sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}} converges to 0, and the results for r (n ) and r ′(n ) depend on theorems in an earlier paper.[ 16]
All the formulas in this section are from Ramanujan's 1918 paper.
Generating functions [ edit ] The generating functions of the Ramanujan sums are Dirichlet series :
ζ ( s ) ∑ δ ∣ q μ ( q δ ) δ 1 − s = ∑ n = 1 ∞ c q ( n ) n s {\displaystyle \zeta (s)\sum _{\delta \,\mid \,q}\mu \left({\frac {q}{\delta }}\right)\delta ^{1-s}=\sum _{n=1}^{\infty }{\frac {c_{q}(n)}{n^{s}}}} is a generating function for the sequence cq (1), cq (2), ... where q is kept constant, and
σ r − 1 ( n ) n r − 1 ζ ( r ) = ∑ q = 1 ∞ c q ( n ) q r {\displaystyle {\frac {\sigma _{r-1}(n)}{n^{r-1}\zeta (r)}}=\sum _{q=1}^{\infty }{\frac {c_{q}(n)}{q^{r}}}} is a generating function for the sequence c 1 (n ), c 2 (n ), ... where n is kept constant.
There is also the double Dirichlet series
ζ ( s ) ζ ( r + s − 1 ) ζ ( r ) = ∑ q = 1 ∞ ∑ n = 1 ∞ c q ( n ) q r n s . {\displaystyle {\frac {\zeta (s)\zeta (r+s-1)}{\zeta (r)}}=\sum _{q=1}^{\infty }\sum _{n=1}^{\infty }{\frac {c_{q}(n)}{q^{r}n^{s}}}.} The polynomial with Ramanujan sum's as coefficients can be expressed with cyclotomic polynomial [ 17]
∑ n = 1 q c q ( n ) x n − 1 = ( x q − 1 ) Φ q ′ ( x ) Φ q ( x ) = Φ q ′ ( x ) ∏ d ∣ q d ≠ q Φ d ( x ) {\displaystyle \sum _{n=1}^{q}c_{q}(n)x^{n-1}=(x^{q}-1){\frac {\Phi _{q}'(x)}{\Phi _{q}(x)}}=\Phi _{q}'(x)\prod _{\begin{array}{c}d\mid q\\[-4pt]d\neq q\end{array}}\Phi _{d}(x)} . σk (n ) is the divisor function (i.e. the sum of the k -th powers of the divisors of n , including 1 and n ). σ0 (n ), the number of divisors of n , is usually written d (n ) and σ1 (n ), the sum of the divisors of n , is usually written σ(n ).
If s > 0,
σ s ( n ) = n s ζ ( s + 1 ) ( c 1 ( n ) 1 s + 1 + c 2 ( n ) 2 s + 1 + c 3 ( n ) 3 s + 1 + ⋯ ) σ − s ( n ) = ζ ( s + 1 ) ( c 1 ( n ) 1 s + 1 + c 2 ( n ) 2 s + 1 + c 3 ( n ) 3 s + 1 + ⋯ ) {\displaystyle {\begin{aligned}\sigma _{s}(n)&=n^{s}\zeta (s+1)\left({\frac {c_{1}(n)}{1^{s+1}}}+{\frac {c_{2}(n)}{2^{s+1}}}+{\frac {c_{3}(n)}{3^{s+1}}}+\cdots \right)\\\sigma _{-s}(n)&=\zeta (s+1)\left({\frac {c_{1}(n)}{1^{s+1}}}+{\frac {c_{2}(n)}{2^{s+1}}}+{\frac {c_{3}(n)}{3^{s+1}}}+\cdots \right)\end{aligned}}} Setting s = 1 gives
σ ( n ) = π 2 6 n ( c 1 ( n ) 1 + c 2 ( n ) 4 + c 3 ( n ) 9 + ⋯ ) . {\displaystyle \sigma (n)={\frac {\pi ^{2}}{6}}n\left({\frac {c_{1}(n)}{1}}+{\frac {c_{2}(n)}{4}}+{\frac {c_{3}(n)}{9}}+\cdots \right).} If the Riemann hypothesis is true, and − 1 2 < s < 1 2 , {\displaystyle -{\tfrac {1}{2}}<s<{\tfrac {1}{2}},}
σ s ( n ) = ζ ( 1 − s ) ( c 1 ( n ) 1 1 − s + c 2 ( n ) 2 1 − s + c 3 ( n ) 3 1 − s + ⋯ ) = n s ζ ( 1 + s ) ( c 1 ( n ) 1 1 + s + c 2 ( n ) 2 1 + s + c 3 ( n ) 3 1 + s + ⋯ ) . {\displaystyle \sigma _{s}(n)=\zeta (1-s)\left({\frac {c_{1}(n)}{1^{1-s}}}+{\frac {c_{2}(n)}{2^{1-s}}}+{\frac {c_{3}(n)}{3^{1-s}}}+\cdots \right)=n^{s}\zeta (1+s)\left({\frac {c_{1}(n)}{1^{1+s}}}+{\frac {c_{2}(n)}{2^{1+s}}}+{\frac {c_{3}(n)}{3^{1+s}}}+\cdots \right).} d (n ) = σ0 (n ) is the number of divisors of n , including 1 and n itself.
− d ( n ) = log 1 1 c 1 ( n ) + log 2 2 c 2 ( n ) + log 3 3 c 3 ( n ) + ⋯ − d ( n ) ( 2 γ + log n ) = log 2 1 1 c 1 ( n ) + log 2 2 2 c 2 ( n ) + log 2 3 3 c 3 ( n ) + ⋯ {\displaystyle {\begin{aligned}-d(n)&={\frac {\log 1}{1}}c_{1}(n)+{\frac {\log 2}{2}}c_{2}(n)+{\frac {\log 3}{3}}c_{3}(n)+\cdots \\-d(n)(2\gamma +\log n)&={\frac {\log ^{2}1}{1}}c_{1}(n)+{\frac {\log ^{2}2}{2}}c_{2}(n)+{\frac {\log ^{2}3}{3}}c_{3}(n)+\cdots \end{aligned}}} where γ = 0.5772... is the Euler–Mascheroni constant .
Euler's totient function φ(n ) is the number of positive integers less than n and coprime to n . Ramanujan defines a generalization of it, if
n = p 1 a 1 p 2 a 2 p 3 a 3 ⋯ {\displaystyle n=p_{1}^{a_{1}}p_{2}^{a_{2}}p_{3}^{a_{3}}\cdots } is the prime factorization of n , and s is a complex number, let
φ s ( n ) = n s ( 1 − p 1 − s ) ( 1 − p 2 − s ) ( 1 − p 3 − s ) ⋯ , {\displaystyle \varphi _{s}(n)=n^{s}(1-p_{1}^{-s})(1-p_{2}^{-s})(1-p_{3}^{-s})\cdots ,} so that φ 1 (n ) = φ (n ) is Euler's function.[ 18]
He proves that
μ ( n ) n s φ s ( n ) ζ ( s ) = ∑ ν = 1 ∞ μ ( n ν ) ν s {\displaystyle {\frac {\mu (n)n^{s}}{\varphi _{s}(n)\zeta (s)}}=\sum _{\nu =1}^{\infty }{\frac {\mu (n\nu )}{\nu ^{s}}}} and uses this to show that
φ s ( n ) ζ ( s + 1 ) n s = μ ( 1 ) c 1 ( n ) φ s + 1 ( 1 ) + μ ( 2 ) c 2 ( n ) φ s + 1 ( 2 ) + μ ( 3 ) c 3 ( n ) φ s + 1 ( 3 ) + ⋯ . {\displaystyle {\frac {\varphi _{s}(n)\zeta (s+1)}{n^{s}}}={\frac {\mu (1)c_{1}(n)}{\varphi _{s+1}(1)}}+{\frac {\mu (2)c_{2}(n)}{\varphi _{s+1}(2)}}+{\frac {\mu (3)c_{3}(n)}{\varphi _{s+1}(3)}}+\cdots .} Letting s = 1,
φ ( n ) = 6 π 2 n ( c 1 ( n ) − c 2 ( n ) 2 2 − 1 − c 3 ( n ) 3 2 − 1 − c 5 ( n ) 5 2 − 1 + c 6 ( n ) ( 2 2 − 1 ) ( 3 2 − 1 ) − c 7 ( n ) 7 2 − 1 + c 10 ( n ) ( 2 2 − 1 ) ( 5 2 − 1 ) − ⋯ ) . {\displaystyle \varphi (n)={\frac {6}{\pi ^{2}}}n\left(c_{1}(n)-{\frac {c_{2}(n)}{2^{2}-1}}-{\frac {c_{3}(n)}{3^{2}-1}}-{\frac {c_{5}(n)}{5^{2}-1}}+{\frac {c_{6}(n)}{(2^{2}-1)(3^{2}-1)}}-{\frac {c_{7}(n)}{7^{2}-1}}+{\frac {c_{10}(n)}{(2^{2}-1)(5^{2}-1)}}-\cdots \right).} Note that the constant is the inverse[ 19] of the one in the formula for σ(n ).
Von Mangoldt's function Λ(n ) = 0 unless n = pk is a power of a prime number, in which case it is the natural logarithm log p .
− Λ ( m ) = c m ( 1 ) + 1 2 c m ( 2 ) + 1 3 c m ( 3 ) + ⋯ {\displaystyle -\Lambda (m)=c_{m}(1)+{\frac {1}{2}}c_{m}(2)+{\frac {1}{3}}c_{m}(3)+\cdots } For all n > 0,
0 = c 1 ( n ) + 1 2 c 2 ( n ) + 1 3 c 3 ( n ) + ⋯ . {\displaystyle 0=c_{1}(n)+{\frac {1}{2}}c_{2}(n)+{\frac {1}{3}}c_{3}(n)+\cdots .} This is equivalent to the prime number theorem .[ 20] [ 21]
r 2s (n ) (sums of squares)[ edit ] r 2s (n ) is the number of way of representing n as the sum of 2s squares , counting different orders and signs as different (e.g., r 2 (13) = 8, as 13 = (±2)2 + (±3)2 = (±3)2 + (±2)2 .)
Ramanujan defines a function δ2s (n ) and references a paper[ 22] in which he proved that r 2s (n ) = δ2s (n ) for s = 1, 2, 3, and 4. For s > 4 he shows that δ2s (n ) is a good approximation to r 2s (n ).
s = 1 has a special formula:
δ 2 ( n ) = π ( c 1 ( n ) 1 − c 3 ( n ) 3 + c 5 ( n ) 5 − ⋯ ) . {\displaystyle \delta _{2}(n)=\pi \left({\frac {c_{1}(n)}{1}}-{\frac {c_{3}(n)}{3}}+{\frac {c_{5}(n)}{5}}-\cdots \right).} In the following formulas the signs repeat with a period of 4.
δ 2 s ( n ) = π s n s − 1 ( s − 1 ) ! ( c 1 ( n ) 1 s + c 4 ( n ) 2 s + c 3 ( n ) 3 s + c 8 ( n ) 4 s + c 5 ( n ) 5 s + c 12 ( n ) 6 s + c 7 ( n ) 7 s + c 16 ( n ) 8 s + ⋯ ) s ≡ 0 ( mod 4 ) δ 2 s ( n ) = π s n s − 1 ( s − 1 ) ! ( c 1 ( n ) 1 s − c 4 ( n ) 2 s + c 3 ( n ) 3 s − c 8 ( n ) 4 s + c 5 ( n ) 5 s − c 12 ( n ) 6 s + c 7 ( n ) 7 s − c 16 ( n ) 8 s + ⋯ ) s ≡ 2 ( mod 4 ) δ 2 s ( n ) = π s n s − 1 ( s − 1 ) ! ( c 1 ( n ) 1 s + c 4 ( n ) 2 s − c 3 ( n ) 3 s + c 8 ( n ) 4 s + c 5 ( n ) 5 s + c 12 ( n ) 6 s − c 7 ( n ) 7 s + c 16 ( n ) 8 s + ⋯ ) s ≡ 1 ( mod 4 ) and s > 1 δ 2 s ( n ) = π s n s − 1 ( s − 1 ) ! ( c 1 ( n ) 1 s − c 4 ( n ) 2 s − c 3 ( n ) 3 s − c 8 ( n ) 4 s + c 5 ( n ) 5 s − c 12 ( n ) 6 s − c 7 ( n ) 7 s − c 16 ( n ) 8 s + ⋯ ) s ≡ 3 ( mod 4 ) {\displaystyle {\begin{aligned}\delta _{2s}(n)&={\frac {\pi ^{s}n^{s-1}}{(s-1)!}}\left({\frac {c_{1}(n)}{1^{s}}}+{\frac {c_{4}(n)}{2^{s}}}+{\frac {c_{3}(n)}{3^{s}}}+{\frac {c_{8}(n)}{4^{s}}}+{\frac {c_{5}(n)}{5^{s}}}+{\frac {c_{12}(n)}{6^{s}}}+{\frac {c_{7}(n)}{7^{s}}}+{\frac {c_{16}(n)}{8^{s}}}+\cdots \right)&&s\equiv 0{\pmod {4}}\\[6pt]\delta _{2s}(n)&={\frac {\pi ^{s}n^{s-1}}{(s-1)!}}\left({\frac {c_{1}(n)}{1^{s}}}-{\frac {c_{4}(n)}{2^{s}}}+{\frac {c_{3}(n)}{3^{s}}}-{\frac {c_{8}(n)}{4^{s}}}+{\frac {c_{5}(n)}{5^{s}}}-{\frac {c_{12}(n)}{6^{s}}}+{\frac {c_{7}(n)}{7^{s}}}-{\frac {c_{16}(n)}{8^{s}}}+\cdots \right)&&s\equiv 2{\pmod {4}}\\[6pt]\delta _{2s}(n)&={\frac {\pi ^{s}n^{s-1}}{(s-1)!}}\left({\frac {c_{1}(n)}{1^{s}}}+{\frac {c_{4}(n)}{2^{s}}}-{\frac {c_{3}(n)}{3^{s}}}+{\frac {c_{8}(n)}{4^{s}}}+{\frac {c_{5}(n)}{5^{s}}}+{\frac {c_{12}(n)}{6^{s}}}-{\frac {c_{7}(n)}{7^{s}}}+{\frac {c_{16}(n)}{8^{s}}}+\cdots \right)&&s\equiv 1{\pmod {4}}{\text{ and }}s>1\\[6pt]\delta _{2s}(n)&={\frac {\pi ^{s}n^{s-1}}{(s-1)!}}\left({\frac {c_{1}(n)}{1^{s}}}-{\frac {c_{4}(n)}{2^{s}}}-{\frac {c_{3}(n)}{3^{s}}}-{\frac {c_{8}(n)}{4^{s}}}+{\frac {c_{5}(n)}{5^{s}}}-{\frac {c_{12}(n)}{6^{s}}}-{\frac {c_{7}(n)}{7^{s}}}-{\frac {c_{16}(n)}{8^{s}}}+\cdots \right)&&s\equiv 3{\pmod {4}}\\\end{aligned}}} and therefore,
r 2 ( n ) = π ( c 1 ( n ) 1 − c 3 ( n ) 3 + c 5 ( n ) 5 − c 7 ( n ) 7 + c 11 ( n ) 11 − c 13 ( n ) 13 + c 15 ( n ) 15 − c 17 ( n ) 17 + ⋯ ) r 4 ( n ) = π 2 n ( c 1 ( n ) 1 − c 4 ( n ) 4 + c 3 ( n ) 9 − c 8 ( n ) 16 + c 5 ( n ) 25 − c 12 ( n ) 36 + c 7 ( n ) 49 − c 16 ( n ) 64 + ⋯ ) r 6 ( n ) = π 3 n 2 2 ( c 1 ( n ) 1 − c 4 ( n ) 8 − c 3 ( n ) 27 − c 8 ( n ) 64 + c 5 ( n ) 125 − c 12 ( n ) 216 − c 7 ( n ) 343 − c 16 ( n ) 512 + ⋯ ) r 8 ( n ) = π 4 n 3 6 ( c 1 ( n ) 1 + c 4 ( n ) 16 + c 3 ( n ) 81 + c 8 ( n ) 256 + c 5 ( n ) 625 + c 12 ( n ) 1296 + c 7 ( n ) 2401 + c 16 ( n ) 4096 + ⋯ ) {\displaystyle {\begin{aligned}r_{2}(n)&=\pi \left({\frac {c_{1}(n)}{1}}-{\frac {c_{3}(n)}{3}}+{\frac {c_{5}(n)}{5}}-{\frac {c_{7}(n)}{7}}+{\frac {c_{11}(n)}{11}}-{\frac {c_{13}(n)}{13}}+{\frac {c_{15}(n)}{15}}-{\frac {c_{17}(n)}{17}}+\cdots \right)\\[6pt]r_{4}(n)&=\pi ^{2}n\left({\frac {c_{1}(n)}{1}}-{\frac {c_{4}(n)}{4}}+{\frac {c_{3}(n)}{9}}-{\frac {c_{8}(n)}{16}}+{\frac {c_{5}(n)}{25}}-{\frac {c_{12}(n)}{36}}+{\frac {c_{7}(n)}{49}}-{\frac {c_{16}(n)}{64}}+\cdots \right)\\[6pt]r_{6}(n)&={\frac {\pi ^{3}n^{2}}{2}}\left({\frac {c_{1}(n)}{1}}-{\frac {c_{4}(n)}{8}}-{\frac {c_{3}(n)}{27}}-{\frac {c_{8}(n)}{64}}+{\frac {c_{5}(n)}{125}}-{\frac {c_{12}(n)}{216}}-{\frac {c_{7}(n)}{343}}-{\frac {c_{16}(n)}{512}}+\cdots \right)\\[6pt]r_{8}(n)&={\frac {\pi ^{4}n^{3}}{6}}\left({\frac {c_{1}(n)}{1}}+{\frac {c_{4}(n)}{16}}+{\frac {c_{3}(n)}{81}}+{\frac {c_{8}(n)}{256}}+{\frac {c_{5}(n)}{625}}+{\frac {c_{12}(n)}{1296}}+{\frac {c_{7}(n)}{2401}}+{\frac {c_{16}(n)}{4096}}+\cdots \right)\end{aligned}}} r ′2s (n) (sums of triangles)[ edit ] r 2 s ′ ( n ) {\displaystyle r'_{2s}(n)} is the number of ways n can be represented as the sum of 2s triangular numbers (i.e. the numbers 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, 15, ...; the n -th triangular number is given by the formula n (n + 1)/2.)
The analysis here is similar to that for squares. Ramanujan refers to the same paper as he did for the squares, where he showed that there is a function δ 2 s ′ ( n ) {\displaystyle \delta '_{2s}(n)} such that r 2 s ′ ( n ) = δ 2 s ′ ( n ) {\displaystyle r'_{2s}(n)=\delta '_{2s}(n)} for s = 1, 2, 3, and 4, and that for s > 4, δ 2 s ′ ( n ) {\displaystyle \delta '_{2s}(n)} is a good approximation to r 2 s ′ ( n ) . {\displaystyle r'_{2s}(n).}
Again, s = 1 requires a special formula:
δ 2 ′ ( n ) = π 4 ( c 1 ( 4 n + 1 ) 1 − c 3 ( 4 n + 1 ) 3 + c 5 ( 4 n + 1 ) 5 − c 7 ( 4 n + 1 ) 7 + ⋯ ) . {\displaystyle \delta '_{2}(n)={\frac {\pi }{4}}\left({\frac {c_{1}(4n+1)}{1}}-{\frac {c_{3}(4n+1)}{3}}+{\frac {c_{5}(4n+1)}{5}}-{\frac {c_{7}(4n+1)}{7}}+\cdots \right).} If s is a multiple of 4,
δ 2 s ′ ( n ) = ( π 2 ) s ( s − 1 ) ! ( n + s 4 ) s − 1 ( c 1 ( n + s 4 ) 1 s + c 3 ( n + s 4 ) 3 s + c 5 ( n + s 4 ) 5 s + ⋯ ) s ≡ 0 ( mod 4 ) δ 2 s ′ ( n ) = ( π 2 ) s ( s − 1 ) ! ( n + s 4 ) s − 1 ( c 1 ( 2 n + s 2 ) 1 s + c 3 ( 2 n + s 2 ) 3 s + c 5 ( 2 n + s 2 ) 5 s + ⋯ ) s ≡ 2 ( mod 4 ) δ 2 s ′ ( n ) = ( π 2 ) s ( s − 1 ) ! ( n + s 4 ) s − 1 ( c 1 ( 4 n + s ) 1 s − c 3 ( 4 n + s ) 3 s + c 5 ( 4 n + s ) 5 s − ⋯ ) s ≡ 1 ( mod 2 ) and s > 1 {\displaystyle {\begin{aligned}\delta '_{2s}(n)&={\frac {({\frac {\pi }{2}})^{s}}{(s-1)!}}\left(n+{\frac {s}{4}}\right)^{s-1}\left({\frac {c_{1}(n+{\frac {s}{4}})}{1^{s}}}+{\frac {c_{3}(n+{\frac {s}{4}})}{3^{s}}}+{\frac {c_{5}(n+{\frac {s}{4}})}{5^{s}}}+\cdots \right)&&s\equiv 0{\pmod {4}}\\[6pt]\delta '_{2s}(n)&={\frac {({\frac {\pi }{2}})^{s}}{(s-1)!}}\left(n+{\frac {s}{4}}\right)^{s-1}\left({\frac {c_{1}(2n+{\frac {s}{2}})}{1^{s}}}+{\frac {c_{3}(2n+{\frac {s}{2}})}{3^{s}}}+{\frac {c_{5}(2n+{\frac {s}{2}})}{5^{s}}}+\cdots \right)&&s\equiv 2{\pmod {4}}\\[6pt]\delta '_{2s}(n)&={\frac {({\frac {\pi }{2}})^{s}}{(s-1)!}}\left(n+{\frac {s}{4}}\right)^{s-1}\left({\frac {c_{1}(4n+s)}{1^{s}}}-{\frac {c_{3}(4n+s)}{3^{s}}}+{\frac {c_{5}(4n+s)}{5^{s}}}-\cdots \right)&&s\equiv 1{\pmod {2}}{\text{ and }}s>1\end{aligned}}} Therefore,
r 2 ′ ( n ) = π 4 ( c 1 ( 4 n + 1 ) 1 − c 3 ( 4 n + 1 ) 3 + c 5 ( 4 n + 1 ) 5 − c 7 ( 4 n + 1 ) 7 + ⋯ ) r 4 ′ ( n ) = ( π 2 ) 2 ( n + 1 2 ) ( c 1 ( 2 n + 1 ) 1 + c 3 ( 2 n + 1 ) 9 + c 5 ( 2 n + 1 ) 25 + ⋯ ) r 6 ′ ( n ) = ( π 2 ) 3 2 ( n + 3 4 ) 2 ( c 1 ( 4 n + 3 ) 1 − c 3 ( 4 n + 3 ) 27 + c 5 ( 4 n + 3 ) 125 − ⋯ ) r 8 ′ ( n ) = ( π 2 ) 4 6 ( n + 1 ) 3 ( c 1 ( n + 1 ) 1 + c 3 ( n + 1 ) 81 + c 5 ( n + 1 ) 625 + ⋯ ) {\displaystyle {\begin{aligned}r'_{2}(n)&={\frac {\pi }{4}}\left({\frac {c_{1}(4n+1)}{1}}-{\frac {c_{3}(4n+1)}{3}}+{\frac {c_{5}(4n+1)}{5}}-{\frac {c_{7}(4n+1)}{7}}+\cdots \right)\\[6pt]r'_{4}(n)&=\left({\frac {\pi }{2}}\right)^{2}\left(n+{\frac {1}{2}}\right)\left({\frac {c_{1}(2n+1)}{1}}+{\frac {c_{3}(2n+1)}{9}}+{\frac {c_{5}(2n+1)}{25}}+\cdots \right)\\[6pt]r'_{6}(n)&={\frac {({\frac {\pi }{2}})^{3}}{2}}\left(n+{\frac {3}{4}}\right)^{2}\left({\frac {c_{1}(4n+3)}{1}}-{\frac {c_{3}(4n+3)}{27}}+{\frac {c_{5}(4n+3)}{125}}-\cdots \right)\\[6pt]r'_{8}(n)&={\frac {({\frac {\pi }{2}})^{4}}{6}}(n+1)^{3}\left({\frac {c_{1}(n+1)}{1}}+{\frac {c_{3}(n+1)}{81}}+{\frac {c_{5}(n+1)}{625}}+\cdots \right)\end{aligned}}} Let
T q ( n ) = c q ( 1 ) + c q ( 2 ) + ⋯ + c q ( n ) U q ( n ) = T q ( n ) + 1 2 ϕ ( q ) {\displaystyle {\begin{aligned}T_{q}(n)&=c_{q}(1)+c_{q}(2)+\cdots +c_{q}(n)\\U_{q}(n)&=T_{q}(n)+{\tfrac {1}{2}}\phi (q)\end{aligned}}} Then for s > 1 ,
σ − s ( 1 ) + ⋯ + σ − s ( n ) = ζ ( s + 1 ) ( n + T 2 ( n ) 2 s + 1 + T 3 ( n ) 3 s + 1 + T 4 ( n ) 4 s + 1 + ⋯ ) = ζ ( s + 1 ) ( n + 1 2 + U 2 ( n ) 2 s + 1 + U 3 ( n ) 3 s + 1 + U 4 ( n ) 4 s + 1 + ⋯ ) − 1 2 ζ ( s ) d ( 1 ) + ⋯ + d ( n ) = − T 2 ( n ) log 2 2 − T 3 ( n ) log 3 3 − T 4 ( n ) log 4 4 − ⋯ d ( 1 ) log 1 + ⋯ + d ( n ) log n = − T 2 ( n ) ( 2 γ log 2 − log 2 2 ) 2 − T 3 ( n ) ( 2 γ log 3 − log 2 3 ) 3 − T 4 ( n ) ( 2 γ log 4 − log 2 4 ) 4 − ⋯ r 2 ( 1 ) + ⋯ + r 2 ( n ) = π ( n − T 3 ( n ) 3 + T 5 ( n ) 5 − T 7 ( n ) 7 + ⋯ ) {\displaystyle {\begin{aligned}\sigma _{-s}(1)+\cdots +\sigma _{-s}(n)&=\zeta (s+1)\left(n+{\frac {T_{2}(n)}{2^{s+1}}}+{\frac {T_{3}(n)}{3^{s+1}}}+{\frac {T_{4}(n)}{4^{s+1}}}+\cdots \right)\\&=\zeta (s+1)\left(n+{\tfrac {1}{2}}+{\frac {U_{2}(n)}{2^{s+1}}}+{\frac {U_{3}(n)}{3^{s+1}}}+{\frac {U_{4}(n)}{4^{s+1}}}+\cdots \right)-{\tfrac {1}{2}}\zeta (s)\\d(1)+\cdots +d(n)&=-{\frac {T_{2}(n)\log 2}{2}}-{\frac {T_{3}(n)\log 3}{3}}-{\frac {T_{4}(n)\log 4}{4}}-\cdots \\d(1)\log 1+\cdots +d(n)\log n&=-{\frac {T_{2}(n)(2\gamma \log 2-\log ^{2}2)}{2}}-{\frac {T_{3}(n)(2\gamma \log 3-\log ^{2}3)}{3}}-{\frac {T_{4}(n)(2\gamma \log 4-\log ^{2}4)}{4}}-\cdots \\r_{2}(1)+\cdots +r_{2}(n)&=\pi \left(n-{\frac {T_{3}(n)}{3}}+{\frac {T_{5}(n)}{5}}-{\frac {T_{7}(n)}{7}}+\cdots \right)\end{aligned}}} ^ Ramanujan, On Certain Trigonometric Sums ... These sums are obviously of great interest, and a few of their properties have been discussed already. But, so far as I know, they have never been considered from the point of view which I adopt in this paper; and I believe that all the results which it contains are new.
(Papers , p. 179). In a footnote cites pp. 360–370 of the Dirichlet–Dedekind Vorlesungen über Zahlentheorie , 4th ed. ^ Nathanson, ch. 8. ^ Hardy & Wright, Thms 65, 66 ^ G. H. Hardy, P. V. Seshu Aiyar, & B. M. Wilson, notes to On certain trigonometrical sums ... , Ramanujan, Papers , p. 343 ^ Schwarz & Spilken (1994) p.16 ^ B. Berndt, commentary to On certain trigonometrical sums... , Ramanujan, Papers , p. 371 ^ Knopfmacher, p. 196 ^ Hardy & Wright, p. 243 ^ Tóth, external links, eq. 6 ^ Tóth, external links, eq. 17. ^ Tóth, external links, eq. 8. ^ B. Berndt, commentary to On certain trigonometrical sums... , Ramanujan, Papers , pp. 369–371 ^ Ramanujan, On certain trigonometrical sums... The majority of my formulae are "elementary" in the technical sense of the word — they can (that is to say) be proved by a combination of processes involving only finite algebra and simple general theorems concerning infinite series
(Papers , p. 179) ^ The theory of formal Dirichlet series is discussed in Hardy & Wright, § 17.6 and in Knopfmacher. ^ Knopfmacher, ch. 7, discusses Ramanujan expansions as a type of Fourier expansion in an inner product space which has the c q as an orthogonal basis. ^ Ramanujan, On Certain Arithmetical Functions ^ Nicol, p. 1 ^ This is Jordan's totient function , Js (n ). ^ Cf. Hardy & Wright, Thm. 329, which states that 6 π 2 < σ ( n ) ϕ ( n ) n 2 < 1. {\displaystyle \;{\frac {6}{\pi ^{2}}}<{\frac {\sigma (n)\phi (n)}{n^{2}}}<1.} ^ Hardy, Ramanujan , p. 141 ^ B. Berndt, commentary to On certain trigonometrical sums... , Ramanujan, Papers , p. 371 ^ Ramanujan, On Certain Arithmetical Functions Hardy, G. H. (1999), Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work , Providence RI: AMS / Chelsea, ISBN 978-0-8218-2023-0 Ramanujan, Srinivasa (1918), "On Certain Trigonometric Sums and their Applications in the Theory of Numbers", Transactions of the Cambridge Philosophical Society , 22 (15): 259–276 (pp. 179–199 of his Collected Papers ) Ramanujan, Srinivasa (1916), "On Certain Arithmetical Functions", Transactions of the Cambridge Philosophical Society , 22 (9): 159–184 (pp. 136–163 of his Collected Papers ) Schwarz, Wolfgang; Spilker, Jürgen (1994), Arithmetical Functions. An introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties , London Mathematical Society Lecture Note Series, vol. 184, Cambridge University Press , ISBN 0-521-42725-8 , Zbl 0807.11001