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Ramanujan's sum math function by Srinivasa Ramanujan
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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
Article In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula
c
q
(
n
)
=
∑
1
≤
a
≤
q
(
a
,
q
)
=
1
e
2
π
i
a
q
n
,
{\displaystyle c_{q}(n)=\sum _{1\leq a\leq q \atop (a,q)=1}e^{2\pi i{\tfrac {a}{q}}n},}
where (a, q) = 1 means that a only takes on values coprime to q.
Srinivasa Ramanujan mentioned the sums in a 1918 paper. 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.
cq(n) is always an integer.
Contents Notation For integers a and b,
a
∣
b
{\displaystyle a\mid b}
is read "a divides b" and means that there exists an integer c such that
a
c
=
b
{\displaystyle ac=b}
;
b
a
=
c
.
{\displaystyle {\frac {b}{a}}=c.}
Similarly,
a
∤
b
{\displaystyle a\nmid b}
is read "a does not divide b".
The summation symbol
∑
d
∣
m
f
(
d
)
Trigonometry The following formulas come from the definition, Euler's formula
e
i
x
=
cos
x
+
i
sin
x
,
{\displaystyle e^{ix}=\cos x+i\sin x,}
and elementary trigonometric identities:
c
1
(
n
)
=
1
c
2
(
n
)
=
cos
n
π
c
3
(
Kluyver Let
ζ
q
=
e
2
π
i
q
.
{\displaystyle \zeta _{q}=e^{\frac {2\pi i}{q}}.}
Then ζq is a root of the equation xq − 1 = 0. Each of its powers,
ζ
q
,
ζ
q
2
,
…
,
ζ
q
q
−
1
,
ζ
q
q
=
ζ
q
0
=
1
{\displaystyle \zeta _{q},\zeta _{q}^{2},\ldots ,\zeta _{q}^{q-1},\zeta _{q}^{q}=\zeta _{q}^{0}=1}
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.
If
(
q
,
r
)
=
1
then
c
q
(
n
)
c
r
(
n
)
=
c
q
r
(
n
)
.
{\displaystyle {\mbox{If }}\;(q,r)=1\;{\mbox{ then }}\;c_{q}(n)c_{r}(n)=c_{qr}(n).}
From the definition (or Kluyver's formula) it is straightforward to prove that, if p is a prime number,
c
p
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
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)}
or of the form:
f
(
q
)
=
∑
n
=
1
∞
a
n
c
q
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}}}}
is a generating function for the sequence cq(1), cq(2), ... where q is kept constant, and
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
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
)
+
⋯
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 }
is the prime factorization of n, and s is a complex number, let
φ
s
(
n
)
=
n
s
(
1
−
p
1
−
s
)
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. For m > 1
−
Λ
(
m
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=
c
m
(
1
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+
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 }
Zero For all n > 0,
0
=
c
1
(
n
)
+
1
2
c
2
(
n
)
+
1
3
c
3
(
n
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+
⋯
.
{\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.
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 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
−
r′2s(n) (sums of triangles) r
2
s
′
(
n
)
{\displaystyle r'_{2s}(n)}
is the number of ways n can be represented as the sum of 2s triangular numbers (i.e. the numbers 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, 15, ...; the n-th triangular number is given by the formula 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)}
such that
r
2
s
′
(
n
)
=
δ
2
s
′
Sums Let
T
q
(
n
)
=
c
q
(
1
)
+
c
q
(
2
)
+
⋯
+
c
q
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n
)
U
q
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=
T
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+
1
2
Article from Wikipedia (CC BY-SA 4.0), where it is maintained by volunteer editors.
{\displaystyle \sum _{d\,\mid \,m}f(d)}
means that d goes through all the positive divisors of m, e.g.
{\displaystyle \sum _{d\,\mid \,12}f(d)=f(1)+f(2)+f(3)+f(4)+f(6)+f(12).}
is the greatest common divisor.
is Euler's totient function.
{\displaystyle \zeta (s)}
is the Riemann zeta function.
{\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,...).
is also a root. Therefore, since there are q of them, they are all of the roots. The numbers
{\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
{\displaystyle \zeta _{q}^{n}=1}
is q. The other primitive q-th roots of unity are the numbers
{\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 that the powers of ζq are precisely the primitive roots for all the divisors of q.
Example. Let q = 12. Then
{\displaystyle \zeta _{12},\zeta _{12}^{5},\zeta _{12}^{7},}
{\displaystyle \zeta _{12}^{11}}
are the primitive twelfth roots of unity,
{\displaystyle \zeta _{12}^{2}}
{\displaystyle \zeta _{12}^{10}}
are the primitive sixth roots of unity,
{\displaystyle \zeta _{12}^{3}=i}
{\displaystyle \zeta _{12}^{9}=-i}
are the primitive fourth roots of unity,
{\displaystyle \zeta _{12}^{4}}
{\displaystyle \zeta _{12}^{8}}
are the primitive third roots of unity,
{\displaystyle \zeta _{12}^{6}=-1}
is the primitive second root of unity, and
{\displaystyle \zeta _{12}^{12}=1}
is the primitive first root of unity.
{\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,
{\displaystyle \eta _{q}(n)=\sum _{d\mid q}c_{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
{\displaystyle \eta _{q}(n)={\begin{cases}0&q\nmid n\\q&q\mid n\\\end{cases}}}
and this leads to the formula
{\displaystyle c_{q}(n)=\sum _{d\mid (q,n)}\mu \left({\frac {q}{d}}\right)d,}
published by Kluyver in 1906.
This shows that cq(n) is always an integer. Compare it with the formula
{\displaystyle \phi (q)=\sum _{d\mid q}\mu \left({\frac {q}{d}}\right)d.}
{\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,
{\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
{\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. The equivalence of it and Ramanujan's sum is due to Hölder.
{\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
{\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
{\displaystyle \{c_{1}(n),c_{2}(n),\ldots \}}
{\displaystyle \sum _{n=a}^{a+q-1}c_{q}(n)=0.}
Let m1, m2 > 0, m = lcm(m1, m2). Then Ramanujan's sums satisfy an orthogonality property:
{\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 \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
{\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}c_{n}(k-a)=\mu (n)c_{n}(a),}
{\displaystyle f(q)=\sum _{n=1}^{\infty }a_{n}c_{q}(n)}
where the ak ∈ C, is called a Ramanujan expansion 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).
The expansion of the zero function depends on a result from the analytic theory of prime numbers, namely that the series
{\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.
All the formulas in this section are from Ramanujan's 1918 paper.
{\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
{\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
{\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 {\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 \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
{\displaystyle -{\tfrac {1}{2}}<s<{\tfrac {1}{2}},}
{\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 {\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.
{\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.
{\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
{\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 \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 of the one in the formula for σ(n).
{\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.
{\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}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 r'_{2s}(n)=\delta '_{2s}(n)}
for s = 1, 2, 3, and 4, and that for s > 4,
{\displaystyle \delta '_{2s}(n)}
is a good approximation to
{\displaystyle r'_{2s}(n).}
Again, s = 1 requires a special formula:
{\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 {\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}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}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}\sigma _{-s}(1)+\cdots +\sigma _{-s}(n)&=\zeta (s+1)\left(n+{\frac {T_{2}(n)}{2^{s+1}}}+{\frac {T_{3}(n)}{3^{s+1}}}+{\frac {T_{4}(n)}{4^{s+1}}}+\cdots \right)\\&=\zeta (s+1)\left(n+{\tfrac {1}{2}}+{\frac {U_{2}(n)}{2^{s+1}}}+{\frac {U_{3}(n)}{3^{s+1}}}+{\frac {U_{4}(n)}{4^{s+1}}}+\cdots \right)-{\tfrac {1}{2}}\zeta (s)\\d(1)+\cdots +d(n)&=-{\frac {T_{2}(n)\log 2}{2}}-{\frac {T_{3}(n)\log 3}{3}}-{\frac {T_{4}(n)\log 4}{4}}-\cdots \\d(1)\log 1+\cdots +d(n)\log n&=-{\frac {T_{2}(n)(2\gamma \log 2-\log ^{2}2)}{2}}-{\frac {T_{3}(n)(2\gamma \log 3-\log ^{2}3)}{3}}-{\frac {T_{4}(n)(2\gamma \log 4-\log ^{2}4)}{4}}-\cdots \\r_{2}(1)+\cdots +r_{2}(n)&=\pi \left(n-{\frac {T_{3}(n)}{3}}+{\frac {T_{5}(n)}{5}}-{\frac {T_{7}(n)}{7}}+\cdots \right)\end{aligned}}}
Ramanujan's sum across IJR 
