In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, Ramanujan's sum, usually denoted ''c
q''(''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
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to ''q''.
Srinivasa Ramanujan
Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis ...
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 In number theory, Vinogradov's theorem is a result which 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 weak conjecture, which would imply the existence of such a rep ...
that every sufficiently large odd number is the sum of three
primes
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
.
Notation
For integers ''a'' and ''b'',
is read "''a'' divides ''b''" and means that there is an integer ''c'' such that
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
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
,
is
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
,
is the
Möbius function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
, and
is the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
.
Formulas for ''c''''q''(''n'')
Trigonometry
These formulas come from the definition,
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...
and elementary trigonometric identities.
:
and so on (, , , ,.., ,...). ''c
q''(''n'') is always an integer.
Kluyver
Let
Then is a root of the equation . Each of its powers,
:
is also a root. Therefore, since there are ''q'' of them, they are all of the roots. The numbers
where 1 ≤ ''n'' ≤ ''q'' are called the ''q''-th
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
. is called a primitive ''q''-th root of unity because the smallest value of ''n'' that makes
is ''q''. The other primitive ''q''-th roots of unity are the numbers
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 are precisely the primitive roots for all the divisors of ''q''.
Example. Let ''q'' = 12. Then
:
and
are the primitive twelfth roots of unity,
:
and
are the primitive sixth roots of unity,
:
and
are the primitive fourth roots of unity,
:
and
are the primitive third roots of unity,
:
is the primitive second root of unity, and
:
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
Moebius, Möbius or Mobius may refer to:
People
* August Ferdinand Möbius (1790–1868), German mathematician and astronomer
* Theodor Möbius (1821–1890), German philologist
* Karl Möbius (1825–1908), German zoologist and ecologist
* Paul ...
,
:
It follows from the identity ''x''
''q'' − 1 = (''x'' − 1)(''x''
''q''−1 + ''x''
''q''−2 + ... + ''x'' + 1) that
:
and this leads to the formula
:
published by Kluyver in 1906.
This shows that ''c''
''q''(''n'') is always an integer. Compare it with the formula
:
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'':
[Schwarz & Spilken (1994) p.16] i.e.
:
From the definition (or Kluyver's formula) it is straightforward to prove that, if ''p'' is a prime number,
:
and if ''p''
''k'' 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 ''c''''q''(''n'')
For all positive integers ''q'',
:
For a fixed value of ''q'' the absolute value of the sequence
is bounded by φ(''q''), and for a fixed value of ''n'' the absolute value of the sequence
is bounded by ''n''.
If ''q'' > 1
:
Let ''m''
1, ''m''
2 > 0, ''m'' = lcm(''m''
1, ''m''
2). Then Ramanujan's sums satisfy an
orthogonality property:
:
Let ''n'', ''k'' > 0. Then
:
known as the
Brauer Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:-
* Alfred Brauer (1894–1985), German-American mathematician, brother of Richard
* Andreas Brauer (born 1973), German film producer
* Arik ...
-
Rademacher identity.
If ''n'' > 0 and ''a'' is any integer, we also have
:
due to Cohen.
Table
Ramanujan expansions
If ''f''(''n'') is an
arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...
(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 , 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 function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
s of the Ramanujan sums are
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analyti ...
:
:
is a generating function for the sequence ''c
q''(1), ''c
q''(2), ... where ''q'' is kept constant, and
:
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
:
σ''k''(''n'')
σ
''k''(''n'') is the
divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...
(i.e. the sum of the ''k''-th powers of the divisors of ''n'', including 1 and ''n''). σ
0(''n''), the number of divisors of ''n'', is usually written ''d''(''n'') and σ
1(''n''), the sum of the divisors of ''n'', is usually written σ(''n'').
If ''s'' > 0,
:
Setting ''s'' = 1 gives
:
If the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
is true, and