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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 ''cq''(''n''), is a function of two positive integer variables ''q'' and ''n'' defined by the formula : c_q(n) = \sum_ e^, 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'', 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_f(d) means that ''d'' goes through all the positive divisors of ''m'', e.g. :\sum_f(d) = f(1) + f(2) + f(3) + f(4) + f(6) + f(12). (a,\,b) 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 ...
, \phi(n) 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 ...
, \mu(n) 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 \zeta(s) 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 ...
e^= \cos x + i \sin x, and elementary trigonometric identities. :\begin c_1(n) &= 1 \\ c_2(n) &= \cos n\pi \\ c_3(n) &= 2\cos \tfrac23 n\pi \\ c_4(n) &= 2\cos \tfrac12 n\pi \\ c_5(n) &= 2\cos \tfrac25 n\pi + 2\cos \tfrac45 n\pi \\ c_6(n) &= 2\cos \tfrac13 n\pi \\ c_7(n) &= 2\cos \tfrac27 n\pi + 2\cos \tfrac47 n\pi + 2\cos \tfrac67 n\pi \\ c_8(n) &= 2\cos \tfrac14 n\pi + 2\cos \tfrac34 n\pi \\ c_9(n) &= 2\cos \tfrac29 n\pi + 2\cos \tfrac49 n\pi + 2\cos \tfrac89 n\pi \\ c_(n)&= 2\cos \tfrac15 n\pi + 2\cos \tfrac35 n\pi \\ \end and so on (, , , ,.., ,...). ''cq''(''n'') is always an integer.


Kluyver

Let \zeta_q=e^. Then is a root of the equation . Each of its powers, :\zeta_q, \zeta_q^2, \ldots, \zeta_q^, \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 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 \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 ''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 are precisely the primitive roots for all the divisors of ''q''. Example. Let ''q'' = 12. Then :\zeta_, \zeta_^5, \zeta_^7, and \zeta_^ are the primitive twelfth roots of unity, :\zeta_^2 and \zeta_^ are the primitive sixth roots of unity, :\zeta_^3 = i and \zeta_^9 = -i are the primitive fourth roots of unity, :\zeta_^4 and \zeta_^8 are the primitive third roots of unity, :\zeta_^6 = -1 is the primitive second root of unity, and :\zeta_^ = 1 is the primitive first root of unity. Therefore, if :\eta_q(n) = \sum_^q \zeta_q^ is the sum of the ''n''-th powers of all the roots, primitive and imprimitive, :\eta_q(n) = \sum_ c_d(n), and by Möbius inversion, :c_q(n) = \sum_ \mu\left(\fracd\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 0 & q\nmid n\\ q & q\mid n\\ \end and this leads to the formula :c_q(n)=\sum_ \mu\left(\frac\right) d, published by Kluyver in 1906. This shows that ''c''''q''(''n'') is always an integer. Compare it with the formula :\phi(q)=\sum_\mu\left(\frac\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'':Schwarz & Spilken (1994) p.16 i.e. :\mbox \;(q,r) = 1 \;\mbox\; c_q(n)c_r(n)=c_(n). From the definition (or Kluyver's formula) it is straightforward to prove that, if ''p'' is a prime number, : c_p(n) = \begin -1 &\mboxp\nmid n\\ \phi(p)&\mboxp\mid n\\ \end , 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, : c_(n) = \begin 0 &\mboxp^\nmid n\\ -p^ &\mboxp^\mid n \mboxp^k\nmid n\\ \phi(p^k) &\mboxp^k\mid n\\ \end . This result and the multiplicative property can be used to prove :c_q(n)= \mu\left(\frac\right)\frac. 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 c_1(q) &= 1 \\ c_q(1) &= \mu(q) \\ c_q(q) &= \phi(q) \\ c_q(m) &= c_q(n) && \text m \equiv n \pmod q \\ \end 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 :\sum_^ c_q(n)=0. Let ''m''1, ''m''2 > 0, ''m'' = lcm(''m''1, ''m''2). Then Ramanujan's sums satisfy an orthogonality property: :\frac\sum_^m c_(k) c_(k) = \begin \phi(m) & m_1=m_2=m,\\ 0 & \text \end Let ''n'', ''k'' > 0. Then :\sum_\stackrel d\;\frac =\frac, 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 :\sum_\stackrel c_n(k-a) = \mu(n)c_n(a), 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: :f(n)=\sum_^\infty a_q c_q(n) or of the form: :f(q)=\sum_^\infty a_n c_q(n) 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 :\sum_^\infty\frac 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 ...
: : \zeta(s) \sum_ \mu\left(\frac\right) \delta^ = \sum_^\infty \frac is a generating function for the sequence , , ... where is kept constant, and :\frac= \sum_^\infty \frac is a generating function for the sequence , , ... where is kept constant. There is also the double Dirichlet series :\frac= \sum_^\infty \sum_^\infty \frac. 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 ...
:\sum_^q c_q(n) x^ = (x^q - 1) \frac = \Phi_q'(x) \prod_ \Phi_d(x)


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 , :\begin \sigma_s(n) &= n^s \zeta(s+1) \left(\frac+ \frac+ \frac+\cdots\right) \\ \sigma_(n) &=\zeta(s+1)\left(\frac+\frac+\frac+\cdots\right) \end Setting gives :\sigma(n)= \fracn \left(\frac+ \frac+ \frac+ \cdots \right). 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 -\tfrac12 :\sigma_s(n) = \zeta(1-s) \left(\frac+ \frac+ \frac+ \cdots \right) = n^s \zeta(1+s) \left( \frac+ \frac+ \frac+ \cdots \right).


''d''(''n'')

is the number of divisors of , including 1 and itself. :\begin -d(n) &= \fracc_1(n)+ \fracc_2(n)+ \fracc_3(n)+ \cdots \\ -d(n)(2\gamma+\log n) &= \fracc_1(n)+ \fracc_2(n)+ \fracc_3(n)+ \cdots \end where is the
Euler–Mascheroni constant Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...
.


''φ''(''n'')

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 number of positive integers less than and coprime to . Ramanujan defines a generalization of it, if :n=p_1^p_2^p_3^\cdots is the
prime factorization In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a comp ...
of , and is a complex number, let :\varphi_s(n)=n^s(1-p_1^)(1-p_2^)(1-p_3^)\cdots, so that is Euler's function. He proves that : \frac= \sum_^\infty \frac and uses this to show that :\frac=\frac+\frac+\frac+\cdots. Letting , :\varphi(n) = \fracn \left(c_1(n) -\frac -\frac -\frac+\frac - \frac +\frac -\cdots \right). Note that the constant is the inverse of the one in the formula for .


Λ(''n'')

Von Mangoldt's function unless is a power of a prime number, in which case it is the
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
. : -\Lambda(m) = c_m(1)+ \frac c_m(2)+ \frac13c_m(3)+\cdots


Zero

For all , :0= c_1(n)+ \frac12c_2(n)+ \frac13c_3(n)+ \cdots. This is equivalent to the
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by p ...
.


''r''2''s''(''n'') (sums of squares)

is the number of ways of representing as the sum of
squares In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
, counting different orders and signs as different (e.g., , as .) Ramanujan defines a function and references a paperRamanujan, ''On Certain Arithmetical Functions'' in which he proved that for . For he shows that is a good approximation to . has a special formula: : \delta_2(n)= \pi \left(\frac- \frac+ \frac- \cdots \right). In the following formulas the signs repeat with a period of 4. :\begin \delta_(n) &= \frac \left( \frac+ \frac+ \frac+\frac+ \frac+ \frac+ \frac+ \frac+ \cdots \right) && s \equiv 0 \pmod 4 \\ pt\delta_(n) &= \frac \left( \frac- \frac+ \frac- \frac+ \frac- \frac+ \frac- \frac+ \cdots \right) && s \equiv 2 \pmod 4 \\ pt\delta_(n) &= \frac \left( \frac+ \frac- \frac+ \frac+ \frac+ \frac- \frac+ \frac+ \cdots \right) && s \equiv 1 \pmod 4 \text s > 1 \\ pt\delta_(n) &= \frac \left(\frac- \frac- \frac- \frac+ \frac-\frac-\frac-\frac+ \cdots \right) && s \equiv 3 \pmod 4 \\ \end and therefore, :\begin r_2(n) &= \pi \left(\frac- \frac+ \frac- \frac+ \frac-\frac+ \frac - \frac + \cdots \right) \\ ptr_4(n) &= \pi^2 n \left( \frac- \frac+ \frac- \frac+ \frac- \frac+ \frac- \frac+ \cdots \right) \\ ptr_6(n) &= \frac \left( \frac- \frac- \frac- \frac+ \frac- \frac- \frac - \frac+ \cdots \right) \\ ptr_8(n) &= \frac \left(\frac+ \frac+ \frac+ \frac+ \frac+ \frac+ \frac+ \frac+ \cdots \right) \end


''r''′2s(n) (sums of triangles)

r'_(n) is the number of ways can be represented as the sum of
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
s (i.e. the numbers 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, 15, ...; the -th triangular number is given by the formula .) 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'_(n) such that r'_(n) = \delta'_(n) for , and that for , \delta'_(n) is a good approximation to r'_(n). Again, requires a special formula: :\delta'_2(n)= \frac \left(\frac-\frac+ \frac- \frac+ \cdots \right). If is a multiple of 4, :\begin \delta'_(n) &= \frac\left(n+\frac4\right)^ \left( \frac+ \frac+ \frac+ \cdots \right) && s \equiv 0 \pmod 4 \\ pt\delta'_(n) &= \frac\left(n+\frac4\right)^ \left( \frac+ \frac+ \frac+ \cdots \right) && s \equiv 2 \pmod 4 \\ pt\delta'_(n) &= \frac\left(n+\frac4\right)^ \left(\frac- \frac+\frac- \cdots \right) && s \equiv 1 \pmod 2 \text s >1 \end Therefore, :\begin r'_2(n) &= \frac \left(\frac- \frac+ \frac- \frac+ \cdots \right) \\ ptr'_4(n) &= \left(\frac\right)^2\left(n+\frac12\right) \left(\frac+\frac+ \frac+ \cdots \right) \\ ptr'_6(n) &= \frac\left(n+\frac34\right)^2 \left(\frac-\frac+ \frac-\cdots \right)\\ ptr'_8(n) &= \frac(n+1)^3 \left(\frac+ \frac+ \frac+ \cdots \right) \end


Sums

Let :\begin T_q(n) &= c_q(1) + c_q(2) + \cdots + c_q(n) \\ U_q(n) &= T_q(n) + \tfrac12\phi(q) \end Then for , :\begin \sigma_(1) + \cdots + \sigma_(n) &= \zeta(s+1) \left(n+ \frac+ \frac+\frac +\cdots \right) \\ &= \zeta(s+1) \left(n+\tfrac12+ \frac+ \frac+ \frac +\cdots \right)- \tfrac12\zeta(s) \\ d(1)+ \cdots+ d(n) &= - \frac - \frac - \frac - \cdots \\ d(1)\log 1 + \cdots + d(n)\log n &= -\frac -\frac -\frac -\cdots \\ r_2(1)+ \cdots+ r_2(n) &= \pi \left(n -\frac +\frac -\frac +\cdots \right) \end


See also

*
Gaussian period In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of root of unity, roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discre ...
* Kloosterman sum


Notes


References

* * * *. * * (pp. 179–199 of his ''Collected Papers'') * (pp. 136–163 of his ''Collected Papers'') * *


External links

* {{cite journal, first1=László , last1= Tóth , title=Sums of products of Ramanujan sums , journal= Annali dell'universita' di Ferrara , arxiv=1104.1906 , year=2011 , doi=10.1007/s11565-011-0143-3 , volume=58 , pages=183–197, s2cid= 119134250 Number theory Squares in number theory Srinivasa Ramanujan