Ramanujan's sum

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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},} {\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]

Notation

For integers a and b, a ∣ b {\displaystyle a\mid 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.} {\displaystyle {\frac {b}{a}}=c.} Similarly, a ∤ b {\displaystyle a\nmid 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)} {\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).} {\displaystyle \sum _{d\,\mid \,12}f(d)=f(1)+f(2)+f(3)+f(4)+f(6)+f(12).}

( a , b ) {\displaystyle (a,\,b)} {\displaystyle (a,\,b)} is the greatest common divisor,

ϕ ( n ) {\displaystyle \phi (n)} {\displaystyle \phi (n)} is Euler's totient function,

μ ( n ) {\displaystyle \mu (n)} {\displaystyle \mu (n)} is the Möbius function, and

ζ ( s ) {\displaystyle \zeta (s)} {\displaystyle \zeta (s)} is the Riemann zeta function.

Formulas for cq(n)

Trigonometry

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,} {\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}}} {\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.

Kluyver

Let ζ q = e 2 π i q . {\displaystyle \zeta _{q}=e^{\frac {2\pi 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} {\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}} {\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} {\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}} {\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},} {\displaystyle \zeta _{12},\zeta _{12}^{5},\zeta _{12}^{7},} and ζ 12 11 {\displaystyle \zeta _{12}^{11}} {\displaystyle \zeta _{12}^{11}} are the primitive twelfth roots of unity,
ζ 12 2 {\displaystyle \zeta _{12}^{2}} {\displaystyle \zeta _{12}^{2}} and ζ 12 10 {\displaystyle \zeta _{12}^{10}} {\displaystyle \zeta _{12}^{10}} are the primitive sixth roots of unity,
ζ 12 3 = i {\displaystyle \zeta _{12}^{3}=i} {\displaystyle \zeta _{12}^{3}=i} and ζ 12 9 = − i {\displaystyle \zeta _{12}^{9}=-i} {\displaystyle \zeta _{12}^{9}=-i} are the primitive fourth roots of unity,
ζ 12 4 {\displaystyle \zeta _{12}^{4}} {\displaystyle \zeta _{12}^{4}} and ζ 12 8 {\displaystyle \zeta _{12}^{8}} {\displaystyle \zeta _{12}^{8}} are the primitive third roots of unity,
ζ 12 6 = − 1 {\displaystyle \zeta _{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} {\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}} {\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),} {\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).} {\displaystyle c_{q}(n)=\sum _{d\mid q}\mu \left({\frac {q}{d}}\right)\eta _{d}(n).}

It follows from the identity xq − 1 = (x − 1)(xq−1 + xq−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}}} {\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,} {\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 cq(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.} {\displaystyle \phi (q)=\sum _{d\mid q}\mu \left({\frac {q}{d}}\right)d.}

von Sterneck

It is easily shown from the definition that cq(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).} {\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}},} {\displaystyle c_{p}(n)={\begin{cases}-1&{\mbox{ if }}p\nmid n\\\phi (p)&{\mbox{ if }}p\mid n\\\end{cases}},}

and if pk 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}}.} {\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)}}.} {\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 cq(n)

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}}} {\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 \}} {\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 \}} {\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.} {\displaystyle \sum _{n=a}^{a+q-1}c_{q}(n)=0.}

Let m1, m2 > 0, m = lcm(m1, m2). 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}}} {\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)}},} {\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),} {\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}c_{n}(k-a)=\mu (n)c_{n}(a),}

due to Cohen.

Table

Ramanujan sum cs(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 111111111 1111111111 1111111111
2 −1 1 −11−11−11−11 −11−11−11−11−11 −11−11−11−11−11
3 −1−1 2 −1−12 −1−12−1−12 −1−12−1−12 −1−12−1−12 −1−12−1−12
4 0−202 0−202 0−202 0−202 0−202 0−202 0−202 0−2
5 −1−1−1−14 −1−1−1−14 −1−1−1−14 −1−1−1−14 −1−1−1−14 −1−1−1−14
6 1−1−2−112 1−1−2−112 1−1−2−112 1−1−2−112 1−1−2−112
7 −1−1−1−1−1−16 −1−1−1−1−1−16 −1−1−1−1−1−16 −1−1−1−1−1−16 −1−1
8 000−40004 000−40004 000−40004 000−400
9 00−300−3006 00−300−3006 00−300−3006 00−3
10 1−11−1−4−11−114 1−11−1−4−11−114 1−11−1−4−11−114
11 −1−1−1−1−1−1−1−1−1−110 −1−1−1−1−1−1−1−1−1−110 −1−1−1−1−1−1−1−1
12 020−20−40−20204 020−20−40−20204 020−20−4
13 −1−1−1−1−1−1−1−1−1−1−1−112 −1−1−1−1−1−1−1−1−1−1−1−112 −1−1−1−1
14 1−11−11−1−6−11−11−116 1−11−11−1−6−11−11−116 1−1
15 11−21−4−211−2−41−2118 11−21−4−211−2−41−2118
16 0000000−800000008 0000000−8000000
17 −1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−116 −1−1−1−1−1−1−1−1−1−1−1−1−1
18 00300−300−600−3003006 00300−300−600−3
19 −1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−118 −1−1−1−1−1−1−1−1−1−1−1
20 020−2020−20−8 0−2020−20208 020−2020−20−8
21 11−211−2−61−211−21−6−211−21112 11−211−2−61−2
22 1−11−11−11−11−1−10 −11−11−11−11−1110 1−11−11−11−1
23 −1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−122 −1−1−1−1−1−1−1
24 0004 000−4 000−8 000−4 0004 0008 0004 00
25 0000−5 0000−5 0000−5 0000−5 000020 0000−5
26 1−11−11−11−11−11−1−12 −11−11−11−11−11−1112 1−11−1
27 00000000−9 00000000−9 0000000018 000
28 020−2020−2020−20−12 0−2020−2020−202012 02
29 −1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−128 −1
30 −11214−2−112−4−1−2−11−8 1−1−2−1−421−1−24121−18

Ramanujan expansions

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)} {\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)} {\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}}} {\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

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}}}} {\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}}}} {\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 c1(n), c2(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}}}.} {\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)} {\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)

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}}} {\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).} {\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}},} {\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).} {\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)

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}}} {\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.

φ(n)

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 } {\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 ,} {\displaystyle \varphi _{s}(n)=n^{s}(1-p_{1}^{-s})(1-p_{2}^{-s})(1-p_{3}^{-s})\cdots ,}

so that φ1(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}}}} {\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 .} {\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).} {\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).

Λ(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 } {\displaystyle -\Lambda (m)=c_{m}(1)+{\frac {1}{2}}c_{m}(2)+{\frac {1}{3}}c_{m}(3)+\cdots }

Zero

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 .} {\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]

r2s(n) (sums of squares)

r2s(n) is the number of ways of representing n as the sum of 2s squares, counting different orders and signs as different (e.g., r2(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 r2s(n) = δ2s(n) for s = 1, 2, 3, and 4. For s > 4 he shows that δ2s(n) is a good approximation to r2s(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).} {\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}}} {\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}}} {\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)

r 2 s ′ ( n ) {\displaystyle r'_{2s}(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 n⁠n + 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)} {\displaystyle \delta '_{2s}(n)} such that r 2 s ′ ( n ) = δ 2 s ′ ( n ) {\displaystyle r'_{2s}(n)=\delta '_{2s}(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)} {\displaystyle \delta '_{2s}(n)} is a good approximation to r 2 s ′ ( n ) . {\displaystyle r'_{2s}(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).} {\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}}} {\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}}} {\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}}}

Sums

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}}} {\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}}} {\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}}}

See also

Notes

  1. 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.
  2. Nathanson, ch. 8.
  3. Hardy & Wright, Thms 65, 66
  4. G. H. Hardy, P. V. Seshu Aiyar, & B. M. Wilson, notes to On certain trigonometrical sums ..., Ramanujan, Papers, p. 343
  5. Schwarz & Spilken (1994) p.16
  6. B. Berndt, commentary to On certain trigonometrical sums..., Ramanujan, Papers, p. 371
  7. Knopfmacher, p. 196
  8. Hardy & Wright, p. 243
  9. Tóth, external links, eq. 6
  10. Tóth, external links, eq. 17.
  11. Tóth, external links, eq. 8.
  12. B. Berndt, commentary to On certain trigonometrical sums..., Ramanujan, Papers, pp. 369–371
  13. 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)
  14. The theory of formal Dirichlet series is discussed in Hardy & Wright, § 17.6 and in Knopfmacher.
  15. Knopfmacher, ch. 7, discusses Ramanujan expansions as a type of Fourier expansion in an inner product space which has the cq as an orthogonal basis.
  16. Ramanujan, On Certain Arithmetical Functions
  17. Nicol, p. 1
  18. This is Jordan's totient function, Js(n).
  19. 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.} {\displaystyle \;{\frac {6}{\pi ^{2}}}<{\frac {\sigma (n)\phi (n)}{n^{2}}}<1.}
  20. Hardy, Ramanujan, p. 141
  21. B. Berndt, commentary to On certain trigonometrical sums..., Ramanujan, Papers, p. 371
  22. Ramanujan, On Certain Arithmetical Functions

References

  • 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.
  • Nathanson, Melvyn B. (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics. Vol. 164. Springer-Verlag. Section A.7. ISBN 0-387-94656-X. Zbl 0859.11002..
  • 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.