Ramanujan's sum

In number theory, a branch of mathematics, 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 _{{a=1 \atop (a,q)=1}}^{q}e^{{2\pi i{\tfrac {a}{q}}n}},

where (a, q) = 1 means that a only takes on values coprime to q.

Srinivasa Ramanujan introduced 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 = ac. Similarly, $a\nmid b$ is read "a does not divide b". The summation symbol

\sum_{d\,\mid\,m}f(d)

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

\sum _{{d\,\mid \,12}}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

These formulas come from the definition, Euler's formula $e^{ix}= \cos x + i \sin x,$ and elementary trigonometric identities.

{\begin{aligned}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{aligned}}

and so on ( A000012, A033999, A099837, A176742,.., A100051,...) They show that c_q(n) is always real.

Kluyver

Let $\zeta_q=e^{\frac{2\pi i}{q}}.$ Then $ζ q$ is a root of the equation $x q - 1 = 0$ . Each of its powers ζ_q, ζ_q², ... ζ_q^q = ζ_q⁰ = 1 is also a root. Therefore, since there are q of them, they are all of the roots. The numbers ζ_qⁿ where 1 ≤ n ≤ q are called the q-th roots of unity. $ζ q$ is called a primitive q-th root of unity because the smallest value of n that makes ζ_qⁿ = 1 is q. The other primitive q-th roots of are the numbers ζ_q^a 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 $ζ q$ are precisely the primitive roots for all the divisors of q.

Example. Let q = 12. Then

ζ₁₂, ζ₁₂⁵, ζ₁₂⁷, and ζ₁₂¹¹ are the primitive twelfth roots of unity,

ζ₁₂² and ζ₁₂¹⁰ are the primitive sixth roots of unity,

ζ₁₂³ = i and ζ₁₂⁹ = −i are the primitive fourth roots of unity,

ζ₁₂⁴ and ζ₁₂⁸ are the primitive third roots of unity,

ζ₁₂⁶ = −1 is the primitive second root of unity, and

ζ₁₂¹² = 1 is the primitive first root of unity.

Therefore, if

\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,

\eta_q(n) = \sum_{d\,\mid\, q} c_d(n),

and by Möbius inversion,

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

\eta _{q}(n)={\begin{cases}0&\;{\mbox{ if }}q\nmid n\\q&\;{\mbox{ if }}q\mid n\\\end{cases}}

and this leads to the formula

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

\phi (q)=\sum _{{d\,\mid \,q}}\mu \left({\frac {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.

\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) = \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) = \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)=\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)

For all positive integers q,

c_{1}(q)=1,\;\;c_{q}(1)=\mu (q),\;{\mbox{ and }}\;c_{q}(q)=\phi (q).

\mbox{If } m \equiv n \pmod q \mbox{ then } c_q(m) = c_q(n) .

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

c_q(1), c_q(2), ... is bounded by φ(q), and

for a fixed value of n the absolute value of the sequence

c₁(n), c₂(n), ... is bounded by n.

If q > 1

\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:

{\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]

\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]

\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)
		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
		n
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_{q=1}^\infty a_q c_q(n)

or of the form:

f(q)=\sum_{n=1}^\infty a_n c_q(n)

where the $a k \in C$ , 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

\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:

\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 c_q(1), c_q(2), ... where q is kept constant, and

{\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

{\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}}}.

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,

\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}}}}+\dots \right)

and

\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}}}}+\dots \right).

Setting s = 1 gives

\sigma (n)={\frac {\pi ^{2}}{6}}n\left({\frac {c_{1}(n)}{1}}+{\frac {c_{2}(n)}{4}}+{\frac {c_{3}(n)}{9}}+\dots \right).

If the Riemann hypothesis is true, and $-\tfrac12<s<\tfrac12,$

\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}}}}+\dots \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}}}}+\dots \right).

d(n)

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

{\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)+\dots \\-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}}}\cdots

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

\phi _{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.^[17]

He proves that

{\frac {\mu (n)n^{s}}{\phi _{s}(n)\zeta (s)}}=\sum _{{\nu =1}}^{\infty }{\frac {\mu (n\nu )}{\nu ^{s}}}

and uses this to show that

\frac{\phi_s(n)\zeta(s+1)}{n^s}=\frac{\mu(1)c_1(n)}{\phi_{s+1}(1)}+\frac{\mu(2)c_2(n)}{\phi_{s+1}(2)}+\frac{\mu(3)c_3(n)}{\phi_{s+1}(3)}+\dots.

Letting s = 1,

\phi (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)}}-\dots \right).

Note that the constant is the inverse^[18] of the one in the formula for σ(n).

Λ(n)

Von Mangoldt's function $Λ(n) = 0$ unless n = p^k is a power of a prime number, in which case it is the natural logarithm log p.

-\Lambda (m)=c_{m}(1)+{\frac {1}{2}}c_{m}(2)+{\frac 13}c_{m}(3)+\dots

Zero

For all n > 0,

0=c_{1}(n)+{\frac 12}c_{2}(n)+{\frac 13}c_{3}(n)+\dots .

This is equivalent to the prime number theorem.^[19]^[20]

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^[21] 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({\frac {c_{1}(n)}{1}}-{\frac {c_{3}(n)}{3}}+{\frac {c_{5}(n)}{5}}-\dots \right).

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

If s ≡ 0 (mod 4),

\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}}}+\dots \right)

If s ≡ 2 (mod 4),

\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}}}+\dots \right)

If s ≡ 1 (mod 4) and s > 1,

\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}+ \dots \right)

If s ≡ 3 (mod 4),

\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}+ \dots \right)

and therefore,

{\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)\\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)\\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)\\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′_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′_2s(n) such that r′_2s(n) = δ′_2s(n) for s = 1, 2, 3, and 4, and that for s > 4, δ′_2s(n) is a good approximation to r′_2s(n).

Again, s = 1 requires a special formula:

\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}}+\dots \right).

If s is a multiple of 4,

\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}}}+\dots \right).

If s is twice an odd number,

\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}}}+\dots \right).

If s is an odd number and s > 1,

\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}}}-\dots \right).

Therefore,

{\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}}+\dots \right)\\r'_{4}(n)&=\left({\tfrac {\pi }{2}}\right)^{2}\left(n+{\tfrac 12}\right)\left({\frac {c_{1}(2n+1)}{1}}+{\frac {c_{3}(2n+1)}{9}}+{\frac {c_{5}(2n+1)}{25}}+\dots \right)\\r'_{6}(n)&={\frac {({\tfrac {\pi }{2}})^{3}}{2}}\left(n+{\tfrac 34}\right)^{2}\left({\frac {c_{1}(4n+3)}{1}}-{\frac {c_{3}(4n+3)}{27}}+{\frac {c_{5}(4n+3)}{125}}-\dots \right)\\r'_{8}(n)&={\frac {({\frac 12}\pi )^{4}}{6}}(n+1)^{3}\left({\frac {c_{1}(n+1)}{1}}+{\frac {c_{3}(n+1)}{81}}+{\frac {c_{5}(n+1)}{625}}+\dots \right)\end{aligned}}

Sums

Let

{\begin{aligned}T_{q}(n)&=c_{q}(1)+c_{q}(2)+\dots +c_{q}(n)\\U_{q}(n)&=T_{q}(n)+{\tfrac 12}\phi (q)\end{aligned}}

Then for $s > 1$ ,

{\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}}}}+\dots \right)\\&=\zeta (s+1)\left(n+{\tfrac 12}+{\frac {U_{2}(n)}{2^{{s+1}}}}+{\frac {U_{3}(n)}{3^{{s+1}}}}+{\frac {U_{4}(n)}{4^{{s+1}}}}+\cdots \right)-{\tfrac 12}\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}}

Notes

↑ 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
↑ This is Jordan's totient function, J_s(n).
↑ Cf. Hardy & Wright, Thm. 329, which states that $\;\frac{6}{\pi^2}<\frac{\sigma(n)\phi(n)}{n^2}<1.$
↑ Hardy, Ramanujan, p. 141
↑ B. Berndt, commentary to On certain trigonometrical sums..., Ramanujan, Papers, p. 371
↑ Ramanujan, On Certain Arithmetical Functions

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

Hardy, G. H.; Wright, E. M. (1979) [1938]. An Introduction to the Theory of Numbers (5th ed.). Oxford: Clarendon Press. ISBN 0-19-853171-0. MR 0568909. Zbl 0423.10001.
Knopfmacher, John (1990) [1975], Abstract Analytic Number Theory (2nd ed.), New York: Dover, ISBN 0-486-66344-2, Zbl 0743.11002

Nathanson, Melvyn B. (1996), Additive Number Theory: the Classical Bases, Graduate Texts in Mathematics, 164, Springer-Verlag, Section A.7, ISBN 0-387-94656-X, Zbl 0859.11002 .
Nicol, C. A. (1962). "Some formulas involving Ramanujan sums". Canad. J. Math. 14: 284–286. doi:10.4153/CJM-1962-019-8.

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)

Ramanujan, Srinivasa (2000), Collected Papers, Providence RI: AMS / Chelsea, ISBN 978-0-8218-2076-6

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, 184, Cambridge University Press, ISBN 0-521-42725-8, Zbl 0807.11001

External links

Tóth, László (2011). "Sums of products of Ramanujan sums". arXiv:1104.1906 .

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