Ramanujan's sum
Encyclopedia
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
where (a, q) = 1 means that a only takes on values coprime
to q.
Srinivasa Ramanujan
introduced 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.
is read "a divides b" and means that there is an integer c such that b = ac. Similarly, 
is read "a does not divide b".
The summation symbol
means that d goes through all the positive divisors of m, e.g.
is the greatest common divisor
,
is Euler's totient function
,
is the Möbius function
, and
is the Riemann zeta function.
and elementary trigonometric identities.
and so on They show that cq(n) is always real.
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 that the powers of ζq are precisely the primitive roots for all the divisors of q.
Therefore, if
is the sum of the n th powers of all the roots, primitive and imprimitive,
and by Möbius inversion,
It follows from the identity xq – 1 = (x – 1)(xq–1 + xq–2 + ... + x + 1) that
and this leads to the formula
published by Kluyver in 1906.
This shows that cq(n) is always an integer. Compare it with the formula
when considered as a function of q for a fixed value of n: i.e.
From the definition (or Kluyver's formula) it is straightforward to prove that, if p is a prime number,
and if pk is a prime power where k > 1,
This result and the multiplicative property can be used to prove
This is called von Sterneck's arithmetic function.
The equivalence of it and Ramanujan's sum is due to Hölder.
For a fixed value of q the absolute value of the sequence
for a fixed value of n the absolute value of the sequence
If q > 1
(i.e. a complex-valued function of the integers or natural numbers), then a convergent infinite series
of the form
or of the form
(where the aq are complex numbers),
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
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.
s of the Ramanujan sums are Dirichlet series:
is a generating function for the sequence cq(1), cq(2), ... where q is kept constant, and
is a generating function for the sequence c1(n), c2(n), ... where n is kept constant.
There is also the double Dirichlet series
(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,
and
Setting s = 1 gives
If the Riemann hypothesis
is true, and
and
where γ = 0.5772... is the Euler–Mascheroni constant
.
φ(n) is the number of positive integers less than n and coprime to n.
Ramanujan defines a generalization of it: if
is the prime factorization of n, and s is a complex number, let
so that φ1(n) = φ(n) is Euler's function.
He proves that
and uses this to show that
Letting s = 1,
Note that the constant is the inverse of the one in the formula for σ(n).
Λ(n) is zero unless n = pk is a power of a prime number, in which case it is the natural logarithm log p.
This is equivalent to the prime number theorem
.
, 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:
In the following formulas the signs repeat with a period of 4.
If s ≡ 0 (mod 4),
If s ≡ 2 (mod 4),
If s ≡ 1 (mod 4) and s > 1,
If s ≡ 3 (mod 4),
and therefore,
s (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:
If s is a multiple of 4,
If s is twice an odd number,
If s is an odd number and s > 1,
Therefore,
and
Then if s > 1,
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula
where (a, q) = 1 means that a only takes on values coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
to q.
Srinivasa Ramanujan
Srinivasa Ramanujan
Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...
introduced 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
Vinogradov's theorem
In number theory, Vinogradov's theorem implies that any sufficiently large odd integer can be written as a sum of three prime numbers. It is a weaker form of Goldbach's conjecture, which would imply the existence of such a representation for all odd integers greater than five. It is named after...
that every sufficiently-large odd number is the sum of three primes.
Notation
For integers a and b, is read "a divides b" and means that there is an integer c such that b = ac. Similarly, is read "a does not divide b".The summation symbol
means that d goes through all the positive divisors of m, e.g. is the greatest common divisorGreatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
,
is Euler's totient functionEuler's totient function
In number theory, the totient \varphi of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n In number theory, the totient \varphi(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that...
,
is the Möbius functionMöbius function
The classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...
, and
is the Riemann zeta function.Trigonometric
These formulas come from the definition, Euler's formulaEuler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
and elementary trigonometric identities.and so on They show that cq(n) is always real.
Kluyver
LetThen 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 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
is the sum of the n th powers of all the roots, primitive and imprimitive,
and by Möbius inversion,
It follows from the identity xq – 1 = (x – 1)(xq–1 + xq–2 + ... + x + 1) that
and this leads to the formula
published by Kluyver in 1906.This shows that cq(n) is always an integer. Compare it with the formula
von Sterneck
It is easily shown from the definition that cq(n) is multiplicativeMultiplicative function
In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
when considered as a function of q for a fixed value of n: i.e.
From the definition (or Kluyver's formula) it is straightforward to prove that, if p is a prime number,
and if pk is a prime power where k > 1,
This result and the multiplicative property can be used to prove
This is called von Sterneck's arithmetic function.The equivalence of it and Ramanujan's sum is due to Hölder.
Other properties of cq(n)
For all positive integers q,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
Table
| 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 f(n) is an arithmetic functionArithmetic function
In number theory, an arithmetic function is a real or complex valued function ƒ defined on the set of natural numbers In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers In number theory, an arithmetic (or...
(i.e. a complex-valued function of the integers or natural numbers), then a convergent infinite series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
of the form
or of the form (where the aq are complex numbers),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
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.
Generating functions
The generating functionGenerating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
s of the Ramanujan sums are Dirichlet series:
is a generating function for the sequence cq(1), cq(2), ... where q is kept constant, and
is a generating function for the sequence c1(n), c2(n), ... where n is kept constant.
There is also the double Dirichlet series
σk(n)
σk(n) is the divisor functionDivisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
(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,
and
Setting s = 1 gives
If the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
is true, and
d(n)
d(n) = σ0(n) is the number of divisors of n, including 1 and n itself.and
where γ = 0.5772... is the Euler–Mascheroni constant
Euler–Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
.
φ(n)
Euler's totient functionEuler's totient function
In number theory, the totient \varphi of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n In number theory, the totient \varphi(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that...
φ(n) is the number of positive integers less than n and coprime to n.
Ramanujan defines a generalization of it: if
is the prime factorization of n, and s is a complex number, let so that φ1(n) = φ(n) is Euler's function.He proves that
and uses this to show that
Letting s = 1,
Note that the constant is the inverse of the one in the formula for σ(n).
Λ(n)
Von Mangoldt's functionVon Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt.-Definition:The von Mangoldt function, conventionally written as Λ, is defined as...
Λ(n) is zero unless n = pk is a power of a prime number, in which case it is the natural logarithm log p.
Zero
For all n > 0,This is equivalent to the prime number theorem
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
.
r2s(n) (sums of squares)
r2s(n) is the number of way of representing n as the sum of 2s squaresSquare number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
, 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:
In the following formulas the signs repeat with a period of 4.
If s ≡ 0 (mod 4),
If s ≡ 2 (mod 4),
If s ≡ 1 (mod 4) and s > 1,
If s ≡ 3 (mod 4),
and therefore,
r′2s(n) (sums of triangles)
r′2s(n) is the number of ways n can be represented as the sum of 2s triangular numberTriangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
s (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:
If s is a multiple of 4,
If s is twice an odd number,
If s is an odd number and s > 1,
Therefore,
Sums
Letand
Then if s > 1,
