Hellenica World

Ramanujan's sum

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

\[ c_q(n)= \sum_{a=1\atop (a,q)=1}^q e^{2 \pi i \tfrac{a}{q} n} , \]

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 cq(n)
Trigonometric

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

\[ \begin{align} c_1(n)& = 1\\ c_2(n) &= \cos n\pi\\ c_3(n)&= 2\cos \tfrac23 n\pi\\ c_4(n)&= 2\cos \tfrac12 n\pi\\ c_5(n)&= 2\cos \tfrac25 n\pi + 2\cos \tfrac45 n\pi\\ c_6(n)&= 2\cos \tfrac13 n\pi \\ c_7(n)&= 2\cos \tfrac27 n\pi + 2\cos \tfrac47 n\pi + 2\cos \tfrac67 n\pi \\ c_8(n)&= 2\cos \tfrac14 n\pi + 2\cos \tfrac34 n\pi \\ c_9(n)&= 2\cos \tfrac29 n\pi + 2\cos \tfrac49 n\pi + 2\cos \tfrac89 n\pi \\ c_{10}(n)&= 2\cos \tfrac15 n\pi + 2\cos \tfrac35 n\pi \\ \end{align} \]

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

Let \[ \zeta_q=e^{\frac{2\pi i}{q}}. \]

Then ζq is a root of the equation xq – 1 = 0. Each of its powers ζq, ζq2, ... ζqq = ζq0 = 1 is also a root. Therefore, since there are q of them, they are all of the roots. The numbers ζqn where 1 ≤ n ≤ q are called the qth roots of unity. ζq is called a primitive q th root of unity because the smallest value of n that makes ζqn = 1 is q. The other primitive qth roots of are the numbers ζqa 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.

For 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 cq(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 cq(n) is multiplicative when considered as a function of q for a fixed value of n: 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 pk 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.[5]

The equivalence of it and Ramanujan's sum is due to Hölder.[6][7]
Other properties of cq(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

\[ cq(1), cq(2), ... \] 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 σ(n), the sum of the divisors of n.

If q > 1

\[ \sum_{n=a}^{a+q-1} c_q(n)=0. \]

Let m1, m2 > 0, m = lcm(m1, m2). Then[8] Ramanujan's sums satisfy an orthogonality property:

\[ \frac{1}{m}\sum_{k=1}^m c_{m_1}(k) c_{m_2}(k) = \begin{cases} \phi(m), & \text{if }\;m_1=m_2=m,\\ 0, & \text{otherwise.} \end{cases} \]

Let n, k > 0. Then[9]

\[ \sum_\stackrel{d\mid n}{\gcd(d,k)=1} d\;\frac{\mu(\tfrac{n}{d})}{\phi(d)} = \frac{\mu(n) c_n(k)}{\phi(n)}, \]

known as the Brauer - Rademacher identity.

If n > 0 and a is any integer, we also have[10]

\[ \sum_\stackrel{1\le k\le n}{\gcd(k,n)=1} c_n(k-a) = \mu(n)c_n(a), \]

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

\[ f(n)=\sum_{q=1}^\infty a_q c_n(q) \] (where the aq are complex numbers),

is called a Ramanujan expansion[11] 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).[12][13][14]

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.[15]

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 cq(1), cq(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

\[ \frac{\zeta(s) \zeta(r+s-1)}{\zeta(r)}= \sum_{q=1}^\infty \sum_{n=1}^\infty \frac{c_q(n)}{q^r n^s} . \]

σk(n)

σk(n) is the divisor function (i.e. the sum of the kth 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,

\[ \begin{align} \sigma_s(n) &= \zeta(1-s) \left( \frac{c_1(n)}{1^{1-s}}+ \frac{c_2(n)}{2^{1-s}}+ \frac{c_3(n)}{3^{1-s}}+ \dots \right)\\ &= n^s \zeta(1+s) \left( \frac{c_1(n)}{1^{1+s}}+ \frac{c_2(n)}{2^{1+s}}+ \frac{c_3(n)}{3^{1+s}}+ \dots \right).\\ \end{align} \]

d(n)

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

\[ -d(n)= \frac{\log 1}{1}c_1(n)+ \frac{\log 2}{2}c_2(n)+ \frac{\log 3}{3}c_3(n)+ \dots \]

and

\[ -d(n)(2\gamma+\log n)= \frac{\log^2 1}{1}c_1(n)+ \frac{\log^2 2}{2}c_2(n)+ \frac{\log^2 3}{3}c_3(n)+ \dots \]

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 \[ n=p_1^{a_1}p_2^{a_2}p_3^{a_3}\dots \] is the prime factorization of n, and s is a complex number, let

\[ \phi_s(n)=n^s(1-p_1^{-s})(1-p_2^{-s})(1-p_3^{-s})\dots \], so that φ1(n) = φ(n) is Euler's function.[16]

He proves that

\[ \frac{\mu(n)n^s}{\phi_s(n)\zeta(s)}= \sum_{\nu=1}^\infty \frac{\mu(m\nu)}{\nu^s} \]

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,

\[ \begin{align} \phi(n) = \frac{6}{\pi^2}n \Big( c_1(n) &-\frac{c_2(n)}{2^2-1} -\frac{c_3(n)}{3^2-1} -\frac{c_5(n)}{5^2-1} \\ &+\frac{c_6(n)}{(2^2-1)(3^2-1)} -\frac{c_7(n)}{7^2-1} +\frac{c_{10}(n)}{(2^2-1)(5^2-1)} -\dots \Big).\\ \end{align} \]

Note that the constant is the inverse[17] of the one in the formula for σ(n).
Λ(n)

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

\[ -\Lambda(m) = c_m(1)+ \frac12c_m(2)+ \frac13c_m(3)+ \dots \]

Zero

For all n > 0,

\[ 0= c_1(n)+ \frac12c_2(n)+ \frac13c_3(n)+ \dots. \]

This is equivalent to the prime number theorem.[18][19]
r2s(n) (sums of squares)

r2s(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 δ2s(n) and references a paper[20] 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:

\[ \delta_2(n)= \pi \left( \frac{c_1(n)}{1}- \frac{c_3(n)}{3}+ \frac{c_5(n)}{5}- \dots \right). \]

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,

\[ r_2(n)= \pi \left( \frac{c_1(n)}{1}- \frac{c_3(n)}{3}+ \frac{c_5(n)}{5}- \frac{c_7(n)}{7}+ \frac{c_{11}(n)}{11}- \frac{c_{13}(n)}{13}+ \frac{c_{15}(n)}{15}- \frac{c_{17}(n)}{17}+ \dots \right) \]

\[ r_4 (n)= \pi^2 n \left( \frac{c_1(n)}{1}- \frac{c_4(n)}{4}+ \frac{c_3(n)}{9}- \frac{c_8(n)}{16}+ \frac{c_5(n)}{25}- \frac{c_{12}(n)}{36}+ \frac{c_7(n)}{49}- \frac{c_{16}(n)}{64}+ \dots \right) \]

\[ r_6(n)= \frac{\pi^3 n^2}{2} \left( \frac{c_1(n)}{1}- \frac{c_4(n)}{8}- \frac{c_3(n)}{27}- \frac{c_8(n)}{64}+ \frac{c_5(n)}{125}- \frac{c_{12}(n)}{216}- \frac{c_7(n)}{343}- \frac{c_{16}(n)}{512}+ \dots \right) \]

\[ r_8(n)= \frac{\pi^4 n^3}{6} \left( \frac{c_1(n)}{1}+ \frac{c_4(n)}{16}+ \frac{c_3(n)}{81}+ \frac{c_8(n)}{256}+ \frac{c_5(n)}{625}+ \frac{c_{12}(n)}{1296}+ \frac{c_7(n)}{2401}+ \frac{c_{16}(n)}{4096}+ \dots \right) \]

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 nth 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,

\[ \delta'_{2s}(n)= \frac{(\frac12\pi)^s}{(s-1)!}\left(n+\frac{s}4\right)^{s-1} \left( \frac{c_1(n+\frac{s}4)}{1^s}+ \frac{c_3(n+\frac{s}4)}{3^s}+ \frac{c_5(n+\frac{s}4)}{5^s}+ \dots \right). \]

If s is twice an odd number,

\[ \delta'_{2s}(n)= \frac{(\frac12\pi)^s}{(s-1)!}\left(n+\frac{s}4\right)^{s-1} \left( \frac{c_1(2n+\frac{s}2)}{1^s}+ \frac{c_3(2n+\frac{s}2)}{3^s}+ \frac{c_5(2n+\frac{s}2)}{5^s}+ \dots \right). \]

If s is an odd number and s > 1,

\[ \delta'_{2s}(n)= \frac{(\frac12\pi)^s}{(s-1)!}\left(n+\frac{s}4\right)^{s-1} \left( \frac{c_1(4n+s)}{1^s}- \frac{c_3(4n+s)}{3^s}+ \frac{c_5(4n+s)}{5^s}- \dots \right). \]

Therefore,

\[ r'_2(n)= \frac{\pi}{4} \left( \frac{c_1(4n+1)}{1}- \frac{c_3(4n+1)}{3}+ \frac{c_5(4n+1)}{5}- \frac{c_7(4n+1)}{7}+ \dots \right) \]

\[ r'_4(n)= \left(\tfrac12\pi\right)^2\left(n+\tfrac12\right) \left( \frac{c_1(2n+1)}{1}+ \frac{c_3(2n+1)}{9}+ \frac{c_5(2n+1)}{25}+ \dots \right) \]

\[ r'_6(n)= \frac{(\frac12\pi)^3}{2}\left(n+\tfrac34\right)^2 \left( \frac{c_1(4n+3)}{1}- \frac{c_3(4n+3)}{27}+ \frac{c_5(4n+3)}{125}- \dots \right) \]

\[ r'_8(n)= \frac{(\frac12\pi)^4}{6}(n+1)^3 \left( \frac{c_1(n+1)}{1}+ \frac{c_3(n+1)}{81}+ \frac{c_5(n+1)}{625}+ \dots \right). \]

Sums

Let

\[ T_q(n) = c_q(1) + c_q(2)+ \dots+c_q(n) \]

and

\[ U_q(n) = T_q(n) + \tfrac12\phi(q). \]

Then if s > 1,

\[ \sigma_{-s}(1)+ \sigma_{-s}(2)+ \dots+ \sigma_{-s}(n) \]

\[ = \zeta(s+1) \left( n+ \frac{T_2(n)}{2^{s+1}}+ \frac{T_3(n)}{3^{s+1}}+ \frac{T_4(n)}{4^{s+1}} +\dots \right) \]

\[ = \zeta(s+1) \left( n+\tfrac12+ \frac{U_2(n)}{2^{s+1}}+ \frac{U_3(n)}{3^{s+1}}+ \frac{U_4(n)}{4^{s+1}} +\dots \right)- \tfrac12\zeta(s) , \]

\[ d(1)+ d(2)+ \dots+ d(n) \]

\[ = -\frac{T_2(n)\log2}{2} -\frac{T_3(n)\log3}{3} -\frac{T_4(n)\log4}{4} -\dots , \]

\[ d(1)\log1+ d(2)\log2+ \dots+ d(n)\log n \]

\[ = -\frac{T_2(n)(2\gamma\log2-\log^22)}{2} -\frac{T_3(n)(2\gamma\log3-\log^23)}{3} -\frac{T_4(n)(2\gamma\log4-\log^24)}{4} -\dots , \]

\[ r_2(1)+ r_2(2)+ \dots+ r_2(n) \]

\[ = \pi \left( n -\frac{T_3(n)}{3} +\frac{T_5(n)}{5} -\frac{T_7(n)}{7} +\dots \right) . \]