Ramanujan's sum

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

$c_q(n)=\sum _{\genfrac{}{}{0ex}{}{1\le a\le q}{(a,q)=1}}{e}^{2\pi i\frac{a}{q}n},$

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.

Notation

For integers a and b, $a\mid b$ is read "a divides b" and means that there is an integer c such that $\frac{b}{a}=c.$ 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,

$\varphi (n)$ 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.

$\begin{array}{cc}{c}_1(n)& =1\\ c_2(n)& =\cos n\pi \\ c_3(n)& =2\cos \frac{2}{3}n\pi \\ c_4(n)& =2\cos \frac{1}{2}n\pi \\ c_5(n)& =2\cos \frac{2}{5}n\pi +2\cos \frac{4}{5}n\pi \\ c_6(n)& =2\cos \frac{1}{3}n\pi \\ c_7(n)& =2\cos \frac{2}{7}n\pi +2\cos \frac{4}{7}n\pi +2\cos \frac{6}{7}n\pi \\ c_8(n)& =2\cos \frac{1}{4}n\pi +2\cos \frac{3}{4}n\pi \\ c_9(n)& =2\cos \frac{2}{9}n\pi +2\cos \frac{4}{9}n\pi +2\cos \frac{8}{9}n\pi \\ c_{10}(n)& =2\cos \frac{1}{5}n\pi +2\cos \frac{3}{5}n\pi \\ \end{array}$

and so on (Template:OEIS2C, Template:OEIS2C, Template:OEIS2C, Template:OEIS2C,.., Template:OEIS2C,...). c_q(n) is always an integer.

Kluyver

Let ${\zeta }_q=e^{\frac{2\pi i}{q}}.$ Then Template:Math is a root of the equation Template:Math. Each of its powers,

${\zeta }_q,{\zeta }_q^2,\dots ,{\zeta }_q^{q-1},{\zeta }_q^q={\zeta }_q^0=1$

is also a root. Therefore, since there are q of them, they are all of the roots. The numbers ${\zeta }_q^n$ where 1 ≤ n ≤ q are called the q-th roots of unity. Template:Math is called a primitive q-th root of unity because the smallest value of n that makes ${\zeta }_q^n=1$ is q. The other primitive q-th roots of unity are the numbers ${\zeta }_q^a$ 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 Template:Math are precisely the primitive roots for all the divisors of q.

Example. Let q = 12. Then

${\zeta }_{12},{\zeta }_{12}^5,{\zeta }_{12}^7,$ and ${\zeta }_{12}^{11}$ are the primitive twelfth roots of unity,

${\zeta }_{12}^2$ and ${\zeta }_{12}^{10}$ are the primitive sixth roots of unity,

${\zeta }_{12}^3=i$ and ${\zeta }_{12}^9=-i$ are the primitive fourth roots of unity,

${\zeta }_{12}^4$ and ${\zeta }_{12}^8$ are the primitive third roots of unity,

${\zeta }_{12}^6=-1$ is the primitive second root of unity, and

${\zeta }_{12}^{12}=1$ 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

${\eta }_q(n)=\{\begin{array}{cc}0& q\nmid n\\ q& q\mid n\end{array}$

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.

This shows that c_q(n) is always an integer. Compare it with the formula

$\varphi (q)=\sum _{d\mid q}\mu \left(\frac{q}{d}\right)d.$

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: i.e.

$\text{If }(q,r)=1\text{ then }{c}_q(n)c_r(n)=c_{qr}(n).$

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

$c_p(n)=\{\begin{array}{cc}-1& \text{ if }p\nmid n\\ \varphi (p)& \text{ if }p\mid n\end{array},$

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

$c_{p^k}(n)=\{\begin{array}{cc}0& \text{ if }{p}^{k-1}\nmid n\\ -p^{k-1}& \text{ if }{p}^{k-1}\mid n\text{ and }{p}^k\nmid n\\ \varphi (p^k)& \text{ if }{p}^k\mid n\end{array}.$

This result and the multiplicative property can be used to prove

$c_q(n)=\mu \left(\frac{q}{(q,n)}\right)\frac{\varphi (q)}{\varphi \left(\frac{q}{(q,n)}\right)}.$

This is called von Sterneck's arithmetic function. The equivalence of it and Ramanujan's sum is due to Hölder.

Other properties of c_q(n)

For all positive integers q,

$\begin{array}{cc}{c}_1(q)& =1\\ c_q(1)& =\mu (q)\\ c_q(q)& =\varphi (q)\\ c_q(m)& =c_q(n)& & \text{for }m\equiv n(\mathrm{mod}q)\\ \end{array}$

For a fixed value of q the absolute value of the sequence $\{c_q(1),c_q(2),\dots \}$ is bounded by φ(q), and for a fixed value of n the absolute value of the sequence $\{c_1(n),c_2(n),\dots \}$ 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 Ramanujan's sums satisfy an orthogonality property:

$\frac{1}{m}\sum _{k=1}^m{c}_{m_1}(k)c_{m_2}(k)=\{\begin{array}{cc}\varphi (m)& m_1=m_2=m,\\ 0& \text{otherwise}\end{array}$

Let n, k > 0. Then

$\sum _{\stackrel{d\mid n}{\gcd (d,k)=1}}d\frac{\mu (\frac{n}{d})}{\varphi (d)}=\frac{\mu (n)c_n(k)}{\varphi (n)},$

known as the Brauer - Rademacher identity.

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

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

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 Template:Math is an arithmetic function (i.e. a complex-valued function of the integers or natural numbers), then a convergent infinite series of the form:

$f(n)=\sum _{q=1}^{\mathrm{\infty }}{a}_q{c}_q(n)$

or of the form:

$f(q)=\sum _{n=1}^{\mathrm{\infty }}{a}_n{c}_q(n)$

where the Template:Math, is called a Ramanujan expansion of Template:Math.

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

$\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n}$

converges to 0, and the results for Template:Math and Template:Math depend on theorems in an earlier paper.

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

Generating functions

The generating functions of the Ramanujan sums are Dirichlet series:

$\zeta (s)\sum _{\delta \mid q}\mu \left(\frac{q}{\delta }\right){\delta }^{1-s}=\sum _{n=1}^{\mathrm{\infty }}\frac{c_q(n)}{n^s}$

is a generating function for the sequence Template:Math, Template:Math, ... where Template:Math is kept constant, and

$\frac{{\sigma }_{r-1}(n)}{n^{r-1}\zeta (r)}=\sum _{q=1}^{\mathrm{\infty }}\frac{c_q(n)}{q^r}$

is a generating function for the sequence Template:Math, Template:Math, ... where Template:Math is kept constant.

There is also the double Dirichlet series

$\frac{\zeta (s)\zeta (r+s-1)}{\zeta (r)}=\sum _{q=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}\frac{c_q(n)}{q^r{n}^s}.$

The polynomial with Ramanujan sum's as coefficients can be expressed with cyclotomic polynomial

$\sum _{n=1}^q{c}_q(n)x^{n-1}=(x^q-1)\frac{{{\mathrm{\Phi }}_q}^{\prime }(x)}{{\mathrm{\Phi }}_q(x)}={{\mathrm{\Phi }}_q}^{\prime }(x)\prod _{\begin{array}{c}d\mid q\\ d\ne q\end{array}}{\mathrm{\Phi }}_d(x)$

σ_k(n)

Template:Math is the divisor function (i.e. the sum of the Template:Math-th powers of the divisors of Template:Mvar, including 1 and Template:Mvar). Template:Math, the number of divisors of Template:Mvar, is usually written Template:Math and Template:Math, the sum of the divisors of Template:Mvar, is usually written Template:Math.

If Template:Math,

$\begin{array}{cc}{\sigma }_s(n)& =n^s\zeta (s+1)(\frac{c_1(n)}{1^{s+1}}+\frac{c_2(n)}{2^{s+1}}+\frac{c_3(n)}{3^{s+1}}+\cdots )\\ {\sigma }_{-s}(n)& =\zeta (s+1)(\frac{c_1(n)}{1^{s+1}}+\frac{c_2(n)}{2^{s+1}}+\frac{c_3(n)}{3^{s+1}}+\cdots )\end{array}$

Setting Template:Math gives

$\sigma (n)=\frac{{\pi }^2}{6}n(\frac{c_1(n)}{1}+\frac{c_2(n)}{4}+\frac{c_3(n)}{9}+\cdots ).$

If the Riemann hypothesis is true, and $-\frac{1}{2}<s<\frac{1}{2},$

${\sigma }_s(n)=\zeta (1-s)(\frac{c_1(n)}{1^{1-s}}+\frac{c_2(n)}{2^{1-s}}+\frac{c_3(n)}{3^{1-s}}+\cdots )=n^s\zeta (1+s)(\frac{c_1(n)}{1^{1+s}}+\frac{c_2(n)}{2^{1+s}}+\frac{c_3(n)}{3^{1+s}}+\cdots ).$

d(n)

Template:Math is the number of divisors of Template:Math, including 1 and Template:Math itself.

$\begin{array}{cc}-d(n)& =\frac{\log 1}{1}{c}_1(n)+\frac{\log 2}{2}{c}_2(n)+\frac{\log 3}{3}{c}_3(n)+\cdots \\ -d(n)(2\gamma +\log n)& =\frac{{\log }^{2}1}{1}{c}_1(n)+\frac{{\log }^{2}2}{2}{c}_2(n)+\frac{{\log }^{2}3}{3}{c}_3(n)+\cdots \end{array}$

where Template:Math is the Euler–Mascheroni constant.

φ(n)

Euler's totient function Template:Math is the number of positive integers less than Template:Mvar and coprime to Template:Mvar. 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 Template:Math, and Template:Math 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 Template:Math is Euler's function.

He proves that

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

and uses this to show that

$\frac{{\phi }_s(n)\zeta (s+1)}{n^s}=\frac{\mu (1)c_1(n)}{{\phi }_{s+1}(1)}+\frac{\mu (2)c_2(n)}{{\phi }_{s+1}(2)}+\frac{\mu (3)c_3(n)}{{\phi }_{s+1}(3)}+\cdots .$

Letting Template:Math,

$\phi (n)=\frac{6}{{\pi }^2}n(c_1(n)-\frac{c_2(n)}{2^2-1}-\frac{c_3(n)}{3^2-1}-\frac{c_5(n)}{5^2-1}+\frac{c_6(n)}{(2^2-1)(3^2-1)}-\frac{c_7(n)}{7^2-1}+\frac{c_{10}(n)}{(2^2-1)(5^2-1)}-\cdots ).$

Note that the constant is the inverse of the one in the formula for Template:Math.

Λ(n)

Von Mangoldt's function Template:Math unless Template:Math is a power of a prime number, in which case it is the natural logarithm Template:Math.

$-\mathrm{\Lambda }(m)=c_m(1)+\frac{1}{2}{c}_m(2)+\frac{1}{3}{c}_m(3)+\cdots$

Zero

For all Template:Math,

$0=c_1(n)+\frac{1}{2}{c}_2(n)+\frac{1}{3}{c}_3(n)+\cdots .$

This is equivalent to the prime number theorem.

r_2s(n) (sums of squares)

Template:Math is the number of ways of representing Template:Math as the sum of Template:Math squares, counting different orders and signs as different (e.g., Template:Math, as Template:Math.)

Ramanujan defines a function Template:Math and references a paper in which he proved that Template:Math for Template:Math. For Template:Math he shows that Template:Math is a good approximation to Template:Math.

Template:Math has a special formula:

${\delta }_2(n)=\pi (\frac{c_1(n)}{1}-\frac{c_3(n)}{3}+\frac{c_5(n)}{5}-\cdots ).$

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

$\begin{array}{cccc}{\delta }_{2s}(n)& =\frac{{\pi }^s{n}^{s-1}}{(s-1)!}(\frac{c_1(n)}{1^s}+\frac{c_4(n)}{2^s}+\frac{c_3(n)}{3^s}+\frac{c_8(n)}{4^s}+\frac{c_5(n)}{5^s}+\frac{c_{12}(n)}{6^s}+\frac{c_7(n)}{7^s}+\frac{c_{16}(n)}{8^s}+\cdots )& & s\equiv 0(\mathrm{mod}4)\\ {\delta }_{2s}(n)& =\frac{{\pi }^s{n}^{s-1}}{(s-1)!}(\frac{c_1(n)}{1^s}-\frac{c_4(n)}{2^s}+\frac{c_3(n)}{3^s}-\frac{c_8(n)}{4^s}+\frac{c_5(n)}{5^s}-\frac{c_{12}(n)}{6^s}+\frac{c_7(n)}{7^s}-\frac{c_{16}(n)}{8^s}+\cdots )& & s\equiv 2(\mathrm{mod}4)\\ {\delta }_{2s}(n)& =\frac{{\pi }^s{n}^{s-1}}{(s-1)!}(\frac{c_1(n)}{1^s}+\frac{c_4(n)}{2^s}-\frac{c_3(n)}{3^s}+\frac{c_8(n)}{4^s}+\frac{c_5(n)}{5^s}+\frac{c_{12}(n)}{6^s}-\frac{c_7(n)}{7^s}+\frac{c_{16}(n)}{8^s}+\cdots )& & s\equiv 1(\mathrm{mod}4)\text{ and }s>1\\ {\delta }_{2s}(n)& =\frac{{\pi }^s{n}^{s-1}}{(s-1)!}(\frac{c_1(n)}{1^s}-\frac{c_4(n)}{2^s}-\frac{c_3(n)}{3^s}-\frac{c_8(n)}{4^s}+\frac{c_5(n)}{5^s}-\frac{c_{12}(n)}{6^s}-\frac{c_7(n)}{7^s}-\frac{c_{16}(n)}{8^s}+\cdots )& & s\equiv 3(\mathrm{mod}4)\\ \end{array}$

and therefore,

$\begin{array}{cc}{r}_2(n)& =\pi (\frac{c_1(n)}{1}-\frac{c_3(n)}{3}+\frac{c_5(n)}{5}-\frac{c_7(n)}{7}+\frac{c_{11}(n)}{11}-\frac{c_{13}(n)}{13}+\frac{c_{15}(n)}{15}-\frac{c_{17}(n)}{17}+\cdots )\\ r_4(n)& ={\pi }^{2}n(\frac{c_1(n)}{1}-\frac{c_4(n)}{4}+\frac{c_3(n)}{9}-\frac{c_8(n)}{16}+\frac{c_5(n)}{25}-\frac{c_{12}(n)}{36}+\frac{c_7(n)}{49}-\frac{c_{16}(n)}{64}+\cdots )\\ r_6(n)& =\frac{{\pi }^3{n}^2}{2}(\frac{c_1(n)}{1}-\frac{c_4(n)}{8}-\frac{c_3(n)}{27}-\frac{c_8(n)}{64}+\frac{c_5(n)}{125}-\frac{c_{12}(n)}{216}-\frac{c_7(n)}{343}-\frac{c_{16}(n)}{512}+\cdots )\\ r_8(n)& =\frac{{\pi }^4{n}^3}{6}(\frac{c_1(n)}{1}+\frac{c_4(n)}{16}+\frac{c_3(n)}{81}+\frac{c_8(n)}{256}+\frac{c_5(n)}{625}+\frac{c_{12}(n)}{1296}+\frac{c_7(n)}{2401}+\frac{c_{16}(n)}{4096}+\cdots )\end{array}$

r′_2s(n) (sums of triangles)

$r{\text{'}}_{2s}(n)$ is the number of ways Template:Math can be represented as the sum of Template:Math triangular numbers (i.e. the numbers 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, 15, ...; the Template:Math-th triangular number is given by the formula Template:Math.)

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 $\delta {\text{'}}_{2s}(n)$ such that $r{\text{'}}_{2s}(n)=\delta {\text{'}}_{2s}(n)$ for Template:Math, and that for Template:Math, $\delta {\text{'}}_{2s}(n)$ is a good approximation to $r{\text{'}}_{2s}(n).$

Again, Template:Math requires a special formula:

$\delta {\text{'}}_2(n)=\frac{\pi }{4}(\frac{c_1(4n+1)}{1}-\frac{c_3(4n+1)}{3}+\frac{c_5(4n+1)}{5}-\frac{c_7(4n+1)}{7}+\cdots ).$

If Template:Math is a multiple of 4,

$\begin{array}{cccc}\delta {\text{'}}_{2s}(n)& =\frac{(\frac{\pi }{2}{)}^s}{(s-1)!}{(n+\frac{s}{4})}^{s-1}(\frac{c_1(n+\frac{s}{4})}{1^s}+\frac{c_3(n+\frac{s}{4})}{3^s}+\frac{c_5(n+\frac{s}{4})}{5^s}+\cdots )& & s\equiv 0(\mathrm{mod}4)\\ \delta {\text{'}}_{2s}(n)& =\frac{(\frac{\pi }{2}{)}^s}{(s-1)!}{(n+\frac{s}{4})}^{s-1}(\frac{c_1(2n+\frac{s}{2})}{1^s}+\frac{c_3(2n+\frac{s}{2})}{3^s}+\frac{c_5(2n+\frac{s}{2})}{5^s}+\cdots )& & s\equiv 2(\mathrm{mod}4)\\ \delta {\text{'}}_{2s}(n)& =\frac{(\frac{\pi }{2}{)}^s}{(s-1)!}{(n+\frac{s}{4})}^{s-1}(\frac{c_1(4n+s)}{1^s}-\frac{c_3(4n+s)}{3^s}+\frac{c_5(4n+s)}{5^s}-\cdots )& & s\equiv 1(\mathrm{mod}2)\text{ and }s>1\end{array}$

Therefore,

$\begin{array}{cc}r{\text{'}}_2(n)& =\frac{\pi }{4}(\frac{c_1(4n+1)}{1}-\frac{c_3(4n+1)}{3}+\frac{c_5(4n+1)}{5}-\frac{c_7(4n+1)}{7}+\cdots )\\ r{\text{'}}_4(n)& ={\left(\frac{\pi }{2}\right)}^2(n+\frac{1}{2})(\frac{c_1(2n+1)}{1}+\frac{c_3(2n+1)}{9}+\frac{c_5(2n+1)}{25}+\cdots )\\ r{\text{'}}_6(n)& =\frac{(\frac{\pi }{2}{)}^3}{2}{(n+\frac{3}{4})}^2(\frac{c_1(4n+3)}{1}-\frac{c_3(4n+3)}{27}+\frac{c_5(4n+3)}{125}-\cdots )\\ r{\text{'}}_8(n)& =\frac{(\frac{\pi }{2}{)}^4}{6}(n+1{)}^3(\frac{c_1(n+1)}{1}+\frac{c_3(n+1)}{81}+\frac{c_5(n+1)}{625}+\cdots )\end{array}$

Sums

Let

$\begin{array}{cc}{T}_q(n)& =c_q(1)+c_q(2)+\cdots +c_q(n)\\ U_q(n)& =T_q(n)+\frac{1}{2}\varphi (q)\end{array}$

Then for Template:Math,

$\begin{array}{cc}{\sigma }_{-s}(1)+\cdots +{\sigma }_{-s}(n)& =\zeta (s+1)(n+\frac{T_2(n)}{2^{s+1}}+\frac{T_3(n)}{3^{s+1}}+\frac{T_4(n)}{4^{s+1}}+\cdots )\\ & =\zeta (s+1)(n+\frac{1}{2}+\frac{U_2(n)}{2^{s+1}}+\frac{U_3(n)}{3^{s+1}}+\frac{U_4(n)}{4^{s+1}}+\cdots )-\frac{1}{2}\zeta (s)\\ d(1)+\cdots +d(n)& =-\frac{T_2(n)\log 2}{2}-\frac{T_3(n)\log 3}{3}-\frac{T_4(n)\log 4}{4}-\cdots \\ d(1)\log 1+\cdots +d(n)\log n& =-\frac{T_2(n)(2\gamma \log 2-{\log }^{2}2)}{2}-\frac{T_3(n)(2\gamma \log 3-{\log }^{2}3)}{3}-\frac{T_4(n)(2\gamma \log 4-{\log }^{2}4)}{4}-\cdots \\ r_2(1)+\cdots +r_2(n)& =\pi (n-\frac{T_3(n)}{3}+\frac{T_5(n)}{5}-\frac{T_7(n)}{7}+\cdots )\end{array}$

See also

• Gaussian period
• Kloosterman sum

References

• Template:Cite book

• Template:Hardy and Wright
• Template:Cite book

• Template:Cite book.

• Template:Cite journal

• Template:Cite journal (pp. 179–199 of his Collected Papers)

• Template:Cite journal (pp. 136–163 of his Collected Papers)

• Template:Cite book

• Template:Cite book

External links

• Template:Cite journal