Ramanujan's sum explained

Ramanujan's sum should not be confused with Ramanujan summation.

In number theory, Ramanujan's sum, usually denoted c_q(n), is a function of two positive integer variables q and n defined by the formula

c _{q(n)} = \sum _{1} e ^{2} {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\mid b is read "a divides b" and means that there is an integer c such that (b)/(a) = c. Similarly, a\nmid b is read "a does not divide b". The summation symbol\sum _{d\midm} f(d)

means that d goes through all the positive divisors of m, e.g.

\sum _{d\mid12} f(d) = f( 1 ) + f( 2 ) + f( 3 ) + f( 4 ) + f( 6 ) + f( 12 ). (a,b) is the greatest common divisor,\phi(n) is Euler's totient function,\mu(n) is the Möbius function, and\zeta(s) is the Riemann zeta function.

Formulas for c_q(n)

Trigonometry

e ^{ix} = \cos x + i \sin x, and elementary trigonometric identities.\begin{align} c _{1(n)} & = 1 \\ c _{2(n)} & = \cos n\pi \\ c _{3(n)} & = 2 \cos \tfrac 23 n\pi \\ c _{4(n)} & = 2 \cos \tfrac 12 n\pi \\ c _{5(n)} & = 2 \cos \tfrac 25 n\pi + 2 \cos \tfrac 45 n\pi \\ c _{6(n)} & = 2 \cos \tfrac 13 n\pi \\ c _{7(n)} & = 2 \cos \tfrac 27 n\pi + 2 \cos \tfrac 47 n\pi + 2 \cos \tfrac 67 n\pi \\ c _{8(n)} & = 2 \cos \tfrac 14 n\pi + 2 \cos \tfrac 34 n\pi \\ c _{9(n)} & = 2 \cos \tfrac 29 n\pi + 2 \cos \tfrac 49 n\pi + 2 \cos \tfrac 89 n\pi \\ c _{10} (n)& = 2 \cos \tfrac 15 n\pi + 2 \cos \tfrac 35 n\pi \\ \end{align}

and so on (..,,...). c_q(n) is always an integer.

Kluyver

Let

\zeta ^{(2 \pi i)/(q)} _{q = e} . Then is a root of the equation . Each of its powers,\zeta _{q,} \zeta ^{2 ,} _{q} \ldots, \zeta ^{q -1} _{q} , \zeta ^{q} _{q} = \zeta ^{0} _{q} =1

is also a root. Therefore, since there are q of them, they are all of the roots. The numbers

\zeta ^{n} _{q} where 1 ≤ n ≤ q are called the q-th roots of unity. is called a primitive q-th root of unity because the smallest value of n that makes \zeta ^{n} _{q} =1 is q. The other primitive q-th roots of unity are the numbers \zeta ^{a} _{q} where (a, q) = 1. Therefore, there are φ(q) primitive q-th roots of unity.

Thus, the Ramanujan sum c_q(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 are precisely the primitive roots for all the divisors of q.

Example. Let q = 12. Then

\zeta _{12} , \zeta ^{5 ,} _{12} \zeta ^{7 ,} _{12} and \zeta ^{11} _{12} are the primitive twelfth roots of unity,\zeta ^{2} _{12} and \zeta ^{10} _{12} are the primitive sixth roots of unity,\zeta ^{3} _{12} = i and \zeta ^{9} _{12} = - i are the primitive fourth roots of unity,\zeta ^{4} _{12} and \zeta ^{8} _{12} are the primitive third roots of unity,\zeta ^{6} _{12} = -1 is the primitive second root of unity, and\zeta ^{12} _{12} = 1 is the primitive first root of unity.

Therefore, if

η _{q(n)} = \sum ^{q} _{k =1} \zeta ^{kn} _{q}

is the sum of the n-th powers of all the roots, primitive and imprimitive,

η _{q(n)} = \sum _{d\mid} c _{d(n),}

and by Möbius inversion,

c _{q(n)} = \sum _{d\mid} \mu\left( (q)/(d\right) η) _{d(n).}

It follows from the identity x^q − 1 = (x − 1)(x^q−1 + x^q−2 + ... + x + 1) that

η _{q(n)} = \begin{cases} 0 & q\nmid n\ q & q\mid n\ \end{cases}

and this leads to the formula

c _{q(n)=\sum} _{d\mid} \mu\left( (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

\phi(q) = \sum _{d} \mu\left( (q)/(d) \right) d.

von Sterneck

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} _{qr} (n).

From the definition (or Kluyver's formula) it is straightforward to prove that, if p is a prime number,

c _{p(n)} = \begin{cases} -1 & if p\nmid n\\ \phi(p)& if p\mid n\\ \end{cases} ,

and if p^k is a prime power where k > 1,

c _{p ^{k}} (n) = \begin{cases} 0 & if p ^{k-1} \nmid n\\ - p ^{k-1} & if p ^{k-1} \mid n and p ^{k\nmid} n\\ \phi(p ^{k)} & if p ^{k\mid} n\\ \end{cases} .

This result and the multiplicative property can be used to prove

c _{q(n)=} \mu\left( (q)/(\right) (q, n)) (\phi(q))/(\phi\left( (q)/(\right) (q, n))) .

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)

For all positive integers q,

\begin{align} c _{1(q)} & = 1 \\ c _{q(1)} & = \mu(q) \\ c _{q(q)} & = \phi(q) \\ c _{q(m)} & = c _{q(n)} && for m \equiv n \pmod q \\ \end{align}

For a fixed value of q the absolute value of the sequence

\{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),} \ldots\} is bounded by n.

If q > 1

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

Let m₁, m₂ > 0, m = lcm(m₁, m₂). Then^[9] Ramanujan's sums satisfy an orthogonality property:

(1)/(m) \sum ^{m} _{k =1} c _{m _{1}} (k) c _{m _{2}} (k) = \begin{cases} \phi(m) & m _{1=m} _{2=m,\ 0} & otherwise \end{cases}

Let n, k > 0. Then^[10]

\sum _{\stackrel{d\mid} n}{\gcd(d,k) =1 } d (\mu(\tfrac{n)/(d) )}{\phi(d)} = (\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]

\sum _{\stackrel{1\le} k\le n}{\gcd(k,n) =1 } c _{n(k-a)} = \mu(n)c _{n(a),}

due to Cohen.

Table

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

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) = \sum ^{inf ty} _{q =1} a _{q} c _{q(n)}

or of the form:

f(q) = \sum ^{inf ty} _{n =1} a _{n} c _{q(n)}

where the, 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

\sum ^{inf ty (\mu(n))/(n)} _{n =1}

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:

\zeta(s) \sum _{\delta\midq} \mu\left( (q)/(\delta) \right) \delta ^{1-s} = \sum ^{inf ty} _{n =1} (c _{q(n)})/(n ^{s})

is a generating function for the sequence c_q(1), c_q(2), ... where q is kept constant, and

(\sigma _{r-1} (n))/(n ^{r-1} \zeta(r)) = \sum ^{inf ty} _{q =1} (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

(\zeta(s) \zeta(r + s -1 ))/(\zeta(r)) = \sum ^{inf ty} _{q =1} \sum ^{inf ty} _{n =1} (c _{q(n)})/(q ^{r} n ^{s}) . The polynomial with Ramanujan sum's as coefficients can be expressed with cyclotomic polynomial^[17] \sum ^{q} _{n =1} c _{q(n)} x ^{n-1} = (x ^{q} - 1 ) (\Phi _{q'(x)})/(\Phi _{q(x)}) = \Phi _{q'(x)} \prod _{\begin{array} {c} d \mid q \\[ -4 pt] d ≠ q\end{array}} \Phi _{d(x)} .

σ_k(n)

σ_k(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,

\begin{align} \sigma _{s(n)} & = n ^{s} \zeta(s +1 ) \left( (c _{1(n)})/(1 ^{s+1}) + (c _{2(n)})/(2 ^{s+1}) + (c _{3(n)})/(3 ^{s+1}) + … \right) \\ \sigma _{-s} (n) & = \zeta(s +1 )\left( (c _{1(n)})/(+ 1 ^{s+1}) (c _{2(n)})/(+ 2 ^{s+1}) (c _{3(n)})/(3 ^{s+1}) + … \right) \end{align}

Setting s = 1 gives

\sigma(n) = (\pi ^{2})/(6) n \left( (c _{1(n)})/(1) + (c _{2(n)})/(4) + (c _{3(n)})/(9) + … \right).

If the Riemann hypothesis is true, and

- \tfrac 12 <s<\tfrac 12 , \sigma _{s(n)} = \zeta( 1- s) \left( (c _{1(n)})/(1 ^{1-s}) + (c _{2(n)})/(2 ^{1-s}) + (c _{3(n)})/(3 ^{1-s}) + … \right) = n ^{s} \zeta( 1+ s) \left( (c _{1(n)})/(1 ^{1+s}) + (c _{2(n)})/(2 ^{1+s}) + (c _{3(n)})/(3 ^{1+s}) + … \right).

d(n)

d(n) = σ₀(n) is the number of divisors of n, including 1 and n itself.

\begin{align} - d(n) & = (log1)/(1) c _{1(n)+} (log2)/(2) c _{2(n)+} (log3)/(3) c _{3(n)+} … \\ - d(n)( 2 \gamma +log n) & = (log ^{2} 1)/(1) c _{1(n)+} (log ^{2} 2)/(2) c _{2(n)+} (log ^{2} 3)/(3) c _{3(n)+} … \end{align}

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 ^{a _{1}} _{1} p ^{a _{2}} _{2} p ^{a _{3}} _{3} …

is the prime factorization of n, and s is a complex number, let

\varphi ^{- s} _{1} )( 1- p ^{- s} _{2} )( 1- p ^{- s} _{3} ) … ,

so that φ₁(n) = φ(n) is Euler's function.^[18]

He proves that

(\mu(n)n ^{s})/(\varphi _{s(n)\zeta(s)}) = \sum ^{inf ty} _{\nu =1} (\mu(n\nu))/(\nu ^{s})

and uses this to show that

(\varphi _{s(n)\zeta(s+1)})/(= n ^{s}) (\mu( 1 )c _{1(n)})/(+ \varphi _{s+1} ( 1 )) (\mu( 2 )c _{2(n)})/(+ \varphi _{s+1} ( 2 )) (\mu( 3 )c _{3(n)})/(\varphi _{s+1} ( 3 )) + … .

Letting s = 1,

\varphi(n) = (6)/(\pi ^{2}) n \left(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)}) - … \right).

Note that the constant is the inverse^[19] of the one in the formula for σ(n).

Λ(n)

Von Mangoldt's function unless n = p^k 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)+} (13 c)/(_{m(3)+ …})

Zero

For all n > 0,

0= c _{1(n)+} (12 c)/(_{2(n)+}) (13 c)/(_{3(n)+}) … .

This is equivalent to the prime number theorem.^{[20] [21]}

r_2s(n) (sums of squares)

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., r₂(13) = 8, as 13 = (±2)² + (±3)² = (±3)² + (±2)².)

Ramanujan defines a function δ_2s(n) and references a paper^[22] in which he proved that r_2s(n) = δ_2s(n) for s = 1, 2, 3, and 4. For s > 4 he shows that δ_2s(n) is a good approximation to r_2s(n).

s = 1 has a special formula:

\delta _{2(n)=} \pi \left( (c _{1(n)})/(1) - (c _{3(n)})/(3) + (c _{5(n)})/(5) - … \right).

In the following formulas the signs repeat with a period of 4.

\begin{align} \delta _{2s} (n) & = (\pi ^{s} n ^{s-1})/((s -1 )!) \left( (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}) + … \right) && s \equiv 0 \pmod 4 \\[ 6 pt] \delta _{2s} (n) & = (\pi ^{s} n ^{s-1})/((s -1 )!) \left( (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}) + … \right) && s \equiv 2 \pmod 4 \\[ 6 pt] \delta _{2s} (n) & = (\pi ^{s} n ^{s-1})/((s -1 )!) \left( (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}) + … \right) && s \equiv 1 \pmod 4 and s > 1 \\[ 6 pt] \delta _{2s} (n) & = (\pi ^{s} n ^{s-1})/(\left( (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}) + … \right) && s \equiv 3 \pmod 4 \\ \end{align}

and therefore,

\begin{align} r _{2(n)} & = \pi \left( (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) + … \right) \\[ 6 pt] r _{4(n)} & = \pi ^{2} n \left( (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) + … \right) \\[ 6 pt] r _{6(n)} & = (\pi ^{3} n ^{2})/(2) \left( (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) + … \right) \\[ 6 pt] r _{8(n)} & = (\pi ^{4} n ^{3})/(\left( 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) + … \right) \end{align}

r′_2s(n) (sums of triangles)

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

\delta' _{2s} (n) such that r' _{2s} (n) = \delta' _{2s} (n) for s = 1, 2, 3, and 4, and that for s > 4, \delta' _{2s} (n) is a good approximation to r' _{2s} (n).

Again, s = 1 requires a special formula:

\delta' _{2(n)=} (\pi)/(\left( 4) (c _{1(4n+1)})/(- 1) (c _{3(4n+1)})/(3) + (c _{5(4n+1)})/(5) - (c _{7(4n+1)})/(7) + … \right).

If s is a multiple of 4,

\begin{align} \delta' _{2s} (n) & = (( (\pi)/() ^{s} 2))/(\left(n + (s -1 )!) (s)/(4 \right)) ^{s-1} \left( (c _{1 (n + (s)/(4 ))})/(1 ^{s}) + (c _{3 (n + (s)/(4 ))})/(3 ^{s}) + (c _{5 (n + (s)/(4 ))})/(5 ^{s}) + … \right) && s \equiv 0 \pmod 4 \\[ 6 pt] \delta' _{2s} (n) & = (( (\pi)/() ^{s} 2))/(\left(n + (s -1 )!) (s)/(4 \right)) ^{s-1} \left( (c _{1 ( 2 n + (s)/(2 ))})/(1 ^{s}) + (c _{3 ( 2 n + (s)/(2 ))})/(3 ^{s}) + (c _{5 ( 2 n + (s)/(2 ))})/(5 ^{s}) + … \right) && s \equiv 2 \pmod 4 \\[ 6 pt] \delta' _{2s} (n) & = (( (\pi)/() ^{s} 2))/(\left(n + (s -1 )!) (s)/(4 \right)) ^{s-1} \left( (c _{1(4n+s)})/(1 ^{s}) - (c _{3(4n+s)})/(+ 3 ^{s}) (c _{5(4n+s)})/(5 ^{s}) - … \right) && s \equiv 1 \pmod 2 and s > 1 \end{align}

Therefore,

\begin{align} r' _{2(n)} & = (\pi)/(\left( 4) (c _{1(4n+1)})/(1) - (c _{3(4n+1)})/(3) + (c _{5(4n+1)})/(5) - (c _{7(4n+1)})/(7) + … \right) \\[ 6 pt] r' _{4(n)} & = \left( (\pi)/(2) \right) ^{2 \left(n + (12 \right))/(\left( (c _{1(2n+1)})/(1))} + (c _{3(2n+1)})/(9) + (c _{5(2n+1)})/(25) + … \right) \\[ 6 pt] r' _{6(n)} & = (( (\pi)/() ^{3} 2))/(\left(n + 2) (34 \right))/(\left( ^{2}) (c _{1(4n+3)})/(- 1) (c _{3(4n+3)})/(27) + (c _{5(4n+3)})/(125) - … \right)\\[ 6 pt] r' _{8(n)} & = (( (\pi)/() ^{4} 2))/(6) (n +1 ) ^{3} \left( (c _{1(n+1)})/(1) + (c _{3(n+1)})/(81) + (c _{5(n+1)})/(625) + … \right) \end{align}

Sums

Let

\begin{align} T _{q(n)} & = c _{q(1)} + c _{q(2)} + … + c _{q(n)} \\ U _{q(n)} & = T _{q(n)} + \tfrac 12 \phi(q) \end{align}

Then for,

\begin{align} \sigma _{-s} ( 1 ) + … + \sigma _{-s} (n) & = \zeta(s +1 ) \left(n + (T _{2(n)})/(2 ^{s+1}) + (T _{3(n)})/(+ 3 ^{s+1}) (T _{4(n)})/(4 ^{s+1}) + … \right) \\ & = \zeta(s +1 ) \left(n + \tfrac 12+ (U _{2(n)})/(2 ^{s+1}) + (U _{3(n)})/(3 ^{s+1}) + (U _{4(n)})/(4 ^{s+1}) + … \right) - \tfrac 12 \zeta(s) \\ d( 1 ) + … + d(n) & = - (T _{2(n)log2})/(2) - (T _{3(n)log3})/(3) - (T _{4(n)log4})/(4) - … \\ d( 1 ) log 1 + … + d(n) log n & =- (T ^{22 )} _{2 (n)( 2 \gamma log2-log})/(- 2) (T ^{23 )} _{3 (n)( 2 \gamma log3-log})/(- 3) (T ^{24 )} _{4 (n)( 2 \gamma log4-log})/(4) - … \\ r _{2(1)+} … + r _{2(n)} & = \pi \left(n - (T _{3(n)})/(+ 3) (T _{5(n)})/(- 5) (T _{7(n)})/(7) + … \right) \end{align}

See also

• Gaussian period
• Kloosterman sum

References

• • • • .
• Some formulas involving Ramanujan sums . 1962. Can. J. Math. . 284–286. 10.4153/CJM-1962-019-8 . C. A. . Nicol . 14 . free .
• (pp. 179 - 199 of his Collected Papers)
• (pp. 136 - 163 of his Collected Papers)

External links

• László . Tóth . Sums of products of Ramanujan sums . Annali dell'universita' di Ferrara . 1104.1906 . 2011 . 10.1007/s11565-011-0143-3 . 58 . 183–197. 119134250 .

Notes and References

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 German: 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 c_q as an orthogonal basis.
16. Ramanujan, On Certain Arithmetical Functions
17. Nicol, p. 1
18. This is Jordan's totient function, J_s(n).
19. Cf. Hardy & Wright, Thm. 329, which states that (6)/(< \pi ^{2}) (\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