Ramanujan's sum

:"This article is not about Ramanujan summation."

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

: $c_q(n)=sum_{a=1atop (a,q)=1}^qe^{2 pi i frac{a}{q} n},$

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

Srinivasa Ramanujan introduced the sums in a 1918 paper. [Ramanujan, S. "On Certain Trigonometric Sums and their Applications in the Theory of Numbers", "Transactions of the Cambridge Philosophical Society", 22, No. 15, (1918), pp 259-276. (pp. 179-199 of his "Collected Papers".) He says "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.", and in a footnote references pp. 360-370 of the Dirichlet-Dedekind "Vorlesungen über Zahlentheorie", 4th ed.]

Notation

For integers $a$ and $b,$ $amid b$ is read "a" divides "b" and means that there is an integer $c$ such that $b=ac.$ Similarly, $a
mid b$ 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")

Trigonometric

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

: $c_1(n) = 1$

: $c_2(n)=cos npi$

: $c_3(n)=2cos frac23 npi$

: $c_4(n)=2cos frac12 npi$

: $c_5(n)=2cos frac25 npi + 2cos frac45 npi$

: $c_6(n)=2cos frac13 npi$

: $c_7(n)=2cos frac27 npi + 2cos frac47 npi + 2cos frac67 npi$

: $c_8(n)=2cos frac14 npi + 2cos frac34 npi$

: $c_9(n)=2cos frac29 npi + 2cos frac49 npi + 2cos frac89 npi$

: $c_{10}(n)=2cos frac15 npi + 2cos frac35 npi$

and so on. They show that "c"_"q"("n") is always real.

Kluyver

Let $zeta_q=e^{frac{2pi i}{q.$

Then ζ_"q" is a root of the equation "x"^"q" – 1 = 0. Each of its powers ζ_"q", ζ_"q"², ... ζ_"q"^"q" = ζ_"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" 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_"q"^"n" = 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 that the powers of ζ_"q" 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,

: $c_q(n) = sum_{d,mid,q} muleft(frac{q}d
ight)eta_d(n).$

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

: $eta_q(n) = egin{cases}&0mbox{ if }q
mid n\&qmbox{ if }qmid n\end{cases}$

and this leads to the formula

: $c_q(n)=sum_{d,mid,(q,n)}muleft(frac{q}{d}
ight) d,$ published by Kluyver in 1906. [Ramanujan, "Papers", notes p. 343]

This shows that "c"_"q"("n") is always an integer. Compare it with $phi(q)=sum_{d,mid,q}muleft(frac{q}{d}
ight) d.$

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": 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,

: $c_p(n) = egin{cases}-1 &mbox{ if }p
mid n\phi(p)&mbox{ if }pmid n\end{cases},$

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

: $c_{p^k}(n) = egin{cases}0 &mbox{ if }p^{k-1}
mid n\-p^{k-1} &mbox{ if }p^{k-1}mid n mbox{ and }p^k
mid n\phi(p^k) &mbox{ if }p^kmid n\end{cases}.$

This result and the multiplicative property can be used to prove : $c_q(n)=muleft(frac{q}{(q, n)}
ight)frac{phi(q)}{phileft(frac{q}{(q, n)}
ight)}.$ This is called von Sterneck's arithmetic function. [Ramanujan, "Papers", notes p. 371]

Other properties of "c"_"q"("n")

For all positive integers "q",

: $c_1(q) = 1, mbox{ } c_q(1) =mu(q), mbox{ and } c_q(q) =phi(q).$

: $mbox{If }m equiv n pmod qmbox{ then }c_q(m) = c_q(n).$

For a fixed value of "q" both of the sequences "c"_"q"(1), "c"_"q"(2), ... and "c"₁("q"), "c"₂("q"), ... remain bounded.

If "q" > 1

: $sum_{n=a}^{a+q-1} c_q(n)=0.$

Table

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

: $f(n)=sum_{q=1}^infty a_q c_q(n)$ where the "a"_"q" are complex numbers, or of the form

: $f(q)=sum_{n=1}^infty a_n c_q(n)$ where the "a"_"n" are complex numbers,

is called a Ramanujan expansion [Ramanujan, "Papers", notes pp. 369-371] 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). [Ramanujan, op. cit.; Hardy, ch. IX (pp. 132-160); Hardy & Wright, Thms 292-293 (pp.250-251); Knopfmacher, ch. 7 (pp. 183-216)]

The expansion of the zero function depends on a result from the analytic theory of prime numbers, namely that the series $sum_{n=1}^inftyfrac{mu(n)}{n}$ converges to 0, and the results for "r"("n") and "r"′("n") depend on theorems in an earlier paper. [Ramanujan, S., "On Certain Arithmetical Functions", "Transactions of the Cambridge Philosophical Society", 22 No. 9, (1916), 159-184; "Collected Papers" pp. 136-163]

All the formulas in this section are from Ramanujan's 1918 paper.

Generating functions

The generating functions of the Ramanujan sums are Dirichlet series.

: $zeta(s)sum_{delta,mid,q}muleft(frac{q}{delta}
ight)delta^{1-s} =sum_{n=1}^inftyfrac{c_q(n)}{n^s}$

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

: $frac{sigma_{r-1}(n)}{n^rzeta(r)}=sum_{q=1}^inftyfrac{c_q(n)}{q^{r$

is a generating function for the sequence "c"₁("n"), "c"₂("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,

: $sigma_s(n)=n^szeta(s+1)left(frac{c_1(n)}{1^{s+1+frac{c_2(n)}{2^{s+1+frac{c_3(n)}{3^{s+1+dots
ight)$

and

: $sigma_{-s}(n)=zeta(s+1)left(frac{c_1(n)}{1^{s+1+frac{c_2(n)}{2^{s+1+frac{c_3(n)}{3^{s+1+dots
ight).$

Setting "s" = 1 gives

: $sigma(n)=frac{pi^2}{6}nleft(frac{c_1(n)}{1}+frac{c_2(n)}{4}+frac{c_3(n)}{9}+dots
ight) .$

If the Riemann hypothesis is true, and $- frac12$

: $egin{align}sigma_s(n)&=zeta(1-s)left(frac{c_1(n)}{1^{1-s+frac{c_2(n)}{2^{1-s+frac{c_3(n)}{3^{1-s+dots
ight)\&=n^szeta(1+s)left(frac{c_1(n)}{1^{1+s+frac{c_2(n)}{2^{1+s+frac{c_3(n)}{3^{1+s+dots
ight).\end{align}$

"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

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

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.

He proves that

: $frac{mu(n)n^s}{phi_s(n)zeta(s)}=sum_{
u=1}^infty frac{mu(m
u)}{
u^s}$

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,

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

Note that the constant is the inverse of the one in the formula for σ("n").

Λ("n")

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

: $-Lambda(m) = c_m(1)+frac12c_m(2)+frac13c_m(3)+dots$

Zero

: $0=c_1(n)+frac12c_2(n)+frac13c_3(n)+dots$

This is equivalent to the prime number theorem.

"r"_2"s"("n") (sums of squares)

"r"_2"s"("n") is the number of way of representing "n" as the sum of 2"s" squares, counting different orders and signs as different (e.g., "r"₂(13) = 8, as 13 = (±2)² + (±3)² = (±3)² + (±2)².)

Ramanujan defines a function δ_2"s"("n") and references a paper [Ramanujan, S., "On Certain Arithmetical Functions", "Transactions of the Cambridge Philosophical Society", 22 No. 9, (1916), 159-184; "Collected Papers" pp. 136-163] in which he proved that "r"_2"s"("n") = δ_2"s"("n") for "s" = 1, 2, 3, and 4. For "s" > 4 he shows that δ_2"s"("n") is a good approximation to "r"_2"s"("n").

"s" = 1 has a special formula:

: $delta_2(n)=pileft(frac{c_1(n)}{1}-frac{c_3(n)}{3}+frac{c_5(n)}{5}-dots
ight)$

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

If "s" &equiv; 0 (mod 4),: $delta_{2s}(n)=frac{pi^s n^{s-1{(s-1)!}left(frac{c_1(n)}{1^s}+frac{c_4(n)}{2^s}+ frac{c_3(n)}{3^s}+ frac{c_8(n)}{4^s}+frac{c_5(n)}{5^s}+frac{c_{12}(n)}{6^s}+frac{c_7(n)}{7^s}+frac{c_{16}(n)}{8^s}+dots
ight)$

If "s" &equiv; 2 (mod 4),: $delta_{2s}(n)=frac{pi^s n^{s-1{(s-1)!}left(frac{c_1(n)}{1^s}-frac{c_4(n)}{2^s}+ frac{c_3(n)}{3^s}- frac{c_8(n)}{4^s}+frac{c_5(n)}{5^s}-frac{c_{12}(n)}{6^s}+frac{c_7(n)}{7^s}-frac{c_{16}(n)}{8^s}+dots
ight)$

If "s" &equiv; 1 (mod 4) and "s" > 1,: $delta_{2s}(n)=frac{pi^s n^{s-1{(s-1)!}left(frac{c_1(n)}{1^s}+frac{c_4(n)}{2^s}-frac{c_3(n)}{3^s}+ frac{c_8(n)}{4^s}+frac{c_5(n)}{5^s}+frac{c_{12}(n)}{6^s}-frac{c_7(n)}{7^s}+frac{c_{16}(n)}{8^s}+dots
ight)$

If "s" &equiv; 3 (mod 4),: $delta_{2s}(n)=frac{pi^s n^{s-1{(s-1)!}left(frac{c_1(n)}{1^s}-frac{c_4(n)}{2^s}-frac{c_3(n)}{3^s}- frac{c_8(n)}{4^s}+frac{c_5(n)}{5^s}-frac{c_{12}(n)}{6^s}-frac{c_7(n)}{7^s}-frac{c_{16}(n)}{8^s}+dots
ight)$

and therefore,

: $r_2(n)= pileft(frac{c_1(n)}{1}-frac{c_3(n)}{3}+frac{c_5(n)}{5}-frac{c_7(n)}{7}+frac{c_{11}(n)}{11}-frac{c_{13}(n)}{13}+frac{c_{15}(n)}{15}-frac{c_{17}(n)}{17}+dots
ight)$

: $r_4 (n)=pi^2 nleft(frac{c_1(n)}{1}-frac{c_4(n)}{4}+frac{c_3(n)}{9}- frac{c_8(n)}{16}+frac{c_5(n)}{25}-frac{c_{12}(n)}{36}+frac{c_7(n)}{49}-frac{c_{16}(n)}{64}+dots
ight)$

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

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

"r"′_2"s"("n") (sums of triangles)

"r"′_2"s"("n") is the number of ways "n" can be represented as the sum of 2"s" 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′_2"s"("n") such that "r"′_2"s"("n") = δ′_2"s"("n") for "s" = 1, 2, 3, and 4, and that for "s" > 4, δ′_2"s"("n") is a good approximation to "r"′_2"s"("n").

Again, "s" = 1 requires a special formula:

: $delta'_2(n)=frac{pi}{4}left(frac{c_1(4n+1)}{1}-frac{c_3(4n+1)}{3}+frac{c_5(4n+1)}{5}-frac{c_7(4n+1)}{7}+dots
ight)$

If "s" is a multiple of 4,: $delta'_{2s}(n)=frac{(frac12pi)^s}{(s-1)!}left(n+frac{s}4
ight)^{s-1}left(frac{c_1(n+frac{s}4)}{1^s}+frac{c_3(n+frac{s}4)}{3^s}+frac{c_5(n+frac{s}4)}{5^s}+dots
ight)$

If "s" is twice an odd number,: $delta'_{2s}(n)=frac{(frac12pi)^s}{(s-1)!}left(n+frac{s}4
ight)^{s-1}left(frac{c_1(2n+frac{s}2)}{1^s}+frac{c_3(2n+frac{s}2)}{3^s}+frac{c_5(2n+frac{s}2)}{5^s}+dots
ight)$

If "s" is an odd number and "s" > 1,: $delta'_{2s}(n)=frac{(frac12pi)^s}{(s-1)!}left(n+frac{s}4
ight)^{s-1}left(frac{c_1(4n+s)}{1^s}-frac{c_3(4n+s)}{3^s}+frac{c_5(4n+s)}{5^s}-dots
ight)$

Therefore,

: $r'_2(n)=frac{pi}{4}left(frac{c_1(4n+1)}{1}-frac{c_3(4n+1)}{3}+frac{c_5(4n+1)}{5}-frac{c_7(4n+1)}{7}+dots
ight)$

: $r'_4(n)=left( frac12pi
ight)^2left(n+ frac12
ight)left(frac{c_1(2n+1)}{1}+frac{c_3(2n+1)}{9}+frac{c_5(2n+1)}{25}+dots
ight)$

: $r'_6(n)=frac{(frac12pi)^3}{2}left(n+ frac34
ight)^2left(frac{c_1(4n+3)}{1}-frac{c_3(4n+3)}{27}+frac{c_5(4n+3)}{125}-dots
ight)$

: $r'_8(n)=frac{(frac12pi)^4}{6}(n+1)^3left(frac{c_1(n+1)}{1}+frac{c_3(n+1)}{81}+frac{c_5(n+1)}{625}+dots
ight)$

ums

Let

: $T_q(n) = c_q(1) + c_q(2)+dots+c_q(n)$

and

: $U_q(n) = T_q(n) + frac12phi(q).$

Then if "s" > 1,: $sigma_{-s}(1)+sigma_{-s}(2)+dots+sigma_{-s}(n)$

:: $=zeta(s+1)left(n+frac{T_2(n)}{2^{s+1+frac{T_3(n)}{3^{s+1+frac{T_4(n)}{4^{s+1+dots
ight)$

:: $=zeta(s+1)left(n+ frac12+frac{U_2(n)}{2^{s+1+frac{U_3(n)}{3^{s+1+frac{U_4(n)}{4^{s+1+dots
ight)- frac12zeta(s),$

: $d(1)+d(2)+dots+d(n)$

:: $=-frac{T_2(n)log2}{2}-frac{T_3(n)log3}{3}-frac{T_4(n)log4}{4}-dots,$

: $d(1)log1+d(2)log2+dots+d(n)log n$

:: $=-frac{T_2(n)(2gammalog2-log^22)}{2}-frac{T_3(n)(2gammalog3-log^23)}{3}-frac{T_4(n)(2gammalog4-log^24)}{4}-dots,$

: $r_2(1)+r_2(2)+dots+r_2(n)$

:: $=pileft(n-frac{T_3(n)}{3}+frac{T_5(n)}{5}-frac{T_7(n)}{7}+dots
ight).$

ee also

*Gaussian period
*Kloosterman sum

Notes

References

*citation
last1 = Hardy | first1 = G. H.
title = Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work
publisher = AMS / Chelsea
location = Providence RI
date = 1999
isbn = 978-0821820230

*citation
last1 = Hardy | first1 = G. H.
last2 = Wright | first2 = E. M.
title = An Introduction to the Theory of Numbers (Fifth edition)
publisher = Oxford University Press
location = Oxford
date = 1980
isbn = 978-0198531715

*citation
last1 = Knopfmacher | first1 = John
title = Abstract Analytic Number Theory
publisher = Dover
location = New York
date = 1990
isbn = 0-486-66344-2

* Section A.7.

*citation
last1 = Ramanujan | first1 = Srinivasa
title = Collected Papers
publisher = AMS / Chelsea
location = Providence RI
date = 2000
isbn = 978-0821820766

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