In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, 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 number theory, 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 equiv ...
to ''q''.
Srinivasa Ramanujan
Srinivasa Ramanujan Aiyangar
(22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...
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'',
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), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest co ...
,
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 \mu(n) 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 m ...
, and
is the
Riemann zeta function.
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 ...
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 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 number theory, the theory of group char ...
. 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,
:
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
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number.
For example: , and are prime powers, while
, and are not.
The sequence of prime powers begins:
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 1 ...
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 Bra ...
-
Rademacher identity.
If ''n'' > 0 and ''a'' is any integer, we also have
:
due to Cohen.
Table
Ramanujan expansions
If is an
arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition th ...
(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 .
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 and 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 representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...
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 anal ...
:
:
is a generating function for the sequence , , ... where is kept constant, and
:
is a generating function for the sequence , , ... where is kept constant.
There is also the double Dirichlet series
:
The polynomial with Ramanujan sum's as coefficients can be expressed with
cyclotomic polynomial
In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th prim ...
:
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'' (includi ...
(i.e. the sum of the -th powers of the divisors of , including 1 and ). , the number of divisors of , is usually written and , the sum of the divisors of , is usually written .
If ,
:
Setting 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 pure ...
is true, and