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| {{distinguish|Ramanujan summation}}
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| In [[number theory]], a branch of [[mathematics]], '''Ramanujan's sum''', usually denoted ''c''<sub>''q''</sub>(''n''), is a function of two positive integer variables ''q'' and ''n'' defined by the formula
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| :<math>c_q(n)=
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| \sum_{a=1\atop (a,q)=1}^q
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| e^{2 \pi i \tfrac{a}{q} n}
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| ,
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| </math>
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| where (''a'', ''q'') = 1 means that ''a'' only takes on values [[coprime]] to ''q''.
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| [[Srinivasa Ramanujan]] introduced the sums in a 1918 paper.<ref>Ramanujan, ''On Certain Trigonometric Sums ...'' <blockquote>These sums are obviously of great interest, and a few of their properties have been discussed already. But, so far as I know, they have never been considered from the point of view which I adopt in this paper; and I believe that all the results which it contains are new.</blockquote>(''Papers'', p. 179). In a footnote cites pp. 360–370 of the Dirichlet-Dedekind ''Vorlesungen über Zahlentheorie'', 4th ed.</ref> 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 [[Prime number|primes]].<ref>Nathanson, ch. 8</ref>
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| ==Notation==
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| For integers ''a'' and ''b'', <math>a\mid b</math> is read "''a'' divides ''b''" and means that there is an integer ''c'' such that ''b'' = ''ac''. Similarly, <math>a\nmid b</math> is read "''a'' does not divide ''b''".
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| The summation symbol <math>\sum_{d\,\mid\,m}f(d)</math> means that ''d'' goes through all the positive divisors of ''m'', e.g.
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| :<math>\sum_{d\,\mid\,12}f(d) =
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| f(1) + f(2) + f(3) + f(4) + f(6) + f(12).
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| </math>
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| <math>(a,\,b)\;</math> is the [[greatest common divisor]],
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| <math>\phi(n)\;</math> is [[Euler's totient function]],
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| <math>\mu(n)\;</math> is the [[Möbius function]], and
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| <math>\zeta(s)\;</math> is the [[Riemann zeta function]].
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| ==Formulas for ''c''<sub>''q''</sub>(''n'')==
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| ===Trigonometry===
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| These formulas come from the definition, [[Euler's formula]] <math>e^{ix}= \cos x + i \sin x,</math> and elementary trigonometric identities.
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| :<math>
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| \begin{align}
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| c_1(n)& =
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| 1\\
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| c_2(n) &=
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| \cos n\pi\\
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| c_3(n)&=
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| 2\cos \tfrac23 n\pi\\
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| c_4(n)&=
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| 2\cos \tfrac12 n\pi\\
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| c_5(n)&=
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| 2\cos \tfrac25 n\pi +
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| 2\cos \tfrac45 n\pi\\
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| c_6(n)&=
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| 2\cos \tfrac13 n\pi \\
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| c_7(n)&=
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| 2\cos \tfrac27 n\pi +
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| 2\cos \tfrac47 n\pi +
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| 2\cos \tfrac67 n\pi \\
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| c_8(n)&=
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| 2\cos \tfrac14 n\pi +
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| 2\cos \tfrac34 n\pi \\
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| c_9(n)&=
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| 2\cos \tfrac29 n\pi +
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| 2\cos \tfrac49 n\pi +
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| 2\cos \tfrac89 n\pi \\
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| c_{10}(n)&=
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| 2\cos \tfrac15 n\pi +
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| 2\cos \tfrac35 n\pi \\
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| \end{align}
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| </math>
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| and so on ({{OEIS2C|A000012}}, {{OEIS2C|A033999}}, {{OEIS2C|A099837}}, {{OEIS2C|A176742}},.., {{OEIS2C|A100051}},...) They show that ''c''<sub>''q''</sub>(''n'') is always real.
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| ===Kluyver===
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| Let <math>\zeta_q=e^{\frac{2\pi i}{q}}.</math>
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| Then ζ<sub>''q''</sub> is a root of the equation ''x''<sup>''q''</sup> – 1 = 0. Each of its powers ζ<sub>''q''</sub>, ζ<sub>''q''</sub><sup>2</sup>, ... ζ<sub>''q''</sub><sup>''q''</sup> = ζ<sub>''q''</sub><sup>0</sup> = 1 is also a root. Therefore, since there are ''q'' of them, they are all of the roots. The numbers ζ<sub>''q''</sub><sup>''n''</sup> where 1 ≤ ''n'' ≤ ''q'' are called the ''q''<sup>th</sup> [[roots of unity]]. ζ<sub>''q''</sub> is called a '''primitive''' ''q'' <sup>th</sup> root of unity because the smallest value of ''n'' that makes ζ<sub>''q''</sub><sup>''n''</sup> = 1 is ''q''. The other primitive ''q''<sup>th</sup> roots of are the numbers ζ<sub>''q''</sub><sup>''a''</sup> where (''a'', ''q'') = 1. Therefore, there are φ(''q'') primitive ''q'' <sup>th</sup> roots of unity.
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| Thus, the Ramanujan sum ''c''<sub>''q''</sub>(''n'') is the sum of the ''n'' <sup>th</sup> powers of the primitive ''q'' <sup>th</sup> roots of unity.
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| It is a fact<ref>Hardy & Wright, Thms 65, 66</ref> that the powers of ζ<sub>''q''</sub> are precisely the primitive roots for all the divisors of ''q''.
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| <blockquote>For example, let ''q'' = 12. Then
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| :ζ<sub>12</sub>, ζ<sub>12</sub><sup>5</sup>, ζ<sub>12</sub><sup>7</sup>, and ζ<sub>12</sub><sup>11</sup> are the primitive twelfth roots of unity,
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| :ζ<sub>12</sub><sup>2</sup> and ζ<sub>12</sub><sup>10</sup> are the primitive sixth roots of unity,
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| :ζ<sub>12</sub><sup>3</sup> = ''i'' and ζ<sub>12</sub><sup>9</sup> = −''i'' are the primitive fourth roots of unity,
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| :ζ<sub>12</sub><sup>4</sup> and ζ<sub>12</sub><sup>8</sup> are the primitive third roots of unity,
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| :ζ<sub>12</sub><sup>6</sup> = −1 is the primitive second root of unity, and
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| :ζ<sub>12</sub><sup>12</sup> = 1 is the primitive first root of unity.
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| </blockquote>
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| Therefore, if
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| :<math>\eta_q(n) = \sum_{k=1}^q \zeta_q^{kn}</math>
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| is the sum of the ''n'' <sup>th</sup> powers of all the roots, primitive and imprimitive,
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| :<math>\eta_q(n) = \sum_{d\,\mid\, q} c_d(n),</math>
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| and by [[Möbius inversion]],
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| :<math>c_q(n) = \sum_{d\,\mid\,q} \mu\left(\frac{q}d\right)\eta_d(n).</math>
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| It follows from the identity ''x''<sup>''q''</sup> – 1 = (''x'' – 1)(''x''<sup>''q''–1</sup> + ''x''<sup>''q''–2</sup> + ... + ''x'' + 1) that
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| :<math>
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| \eta_q(n) =
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| \begin{cases}
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| 0&\;\mbox{ if }q\nmid n\\
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| q&\;\mbox{ if }q\mid n\\
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| \end{cases}
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| </math>
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| and this leads to the formula
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| :<math>
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| c_q(n)=
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| \sum_{d\,\mid\,(q,n)}\mu\left(\frac{q}{d}\right) d
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| ,
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| </math> published by Kluyver in 1906.<ref>G. H. Hardy, P. V. Seshu Aiyar, & B. M. Wilson, notes to ''On certain trigonometrical sums ...'', Ramanujan, ''Papers'', p. 343</ref>
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| This shows that ''c''<sub>''q''</sub>(''n'') is always an integer. Compare it with the formula
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| :<math>
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| \phi(q)=
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| \sum_{d\,\mid\,q}\mu\left(\frac{q}{d}\right) d
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| .</math>
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| ===von Sterneck===
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| It is easily shown from the definition that ''c''<sub>''q''</sub>(''n'') is [[multiplicative function|multiplicative]] when considered as a function of ''q'' for a fixed value of ''n'': i.e.
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| :<math>\mbox{If } \;(q,r) = 1 \;\mbox{ then }\; c_q(n)c_r(n)=c_{qr}(n).</math>
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| From the definition (or Kluyver's formula) it is straightforward to prove that, if ''p'' is a prime number,
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| :<math>
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| c_p(n) =
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| \begin{cases}
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| -1 &\mbox{ if }p\nmid n\\
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| \phi(p)&\mbox{ if }p\mid n\\
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| \end{cases}
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| ,</math>
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| and if ''p''<sup>''k''</sup> is a prime power where ''k'' > 1,
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| :<math>
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| c_{p^k}(n) =
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| \begin{cases}
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| 0 &\mbox{ if }p^{k-1}\nmid n\\
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| -p^{k-1} &\mbox{ if }p^{k-1}\mid n \mbox{ and }p^k\nmid n\\
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| \phi(p^k) &\mbox{ if }p^k\mid n\\
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| \end{cases}
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| .</math> | |
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| This result and the multiplicative property can be used to prove
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| :<math>c_q(n)=
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| \mu\left(\frac{q}{(q, n)}\right)
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| \frac{\phi(q)}{\phi\left(\frac{q}{(q, n)}\right)}
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| .
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| </math> This is called von Sterneck's arithmetic function.<ref>B. Berndt, commentary to ''On certain trigonometrical sums...'', Ramanujan, ''Papers'', p. 371</ref>
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| The equivalence of it and Ramanujan's sum is due to Hölder.<ref>Knopfmacher, p. 196</ref><ref>Hardy & Wright, p. 243</ref>
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| ===Other properties of ''c''<sub>''q''</sub>(''n'')===
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| For all positive integers ''q'',
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| :<math>
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| c_1(q) = 1, \;\;
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| c_q(1) = \mu(q), \;
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| \mbox{ and }\; c_q(q) =
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| \phi(q)
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| .
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| </math>
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| :<math>
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| \mbox{If }
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| m \equiv n \pmod q
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| \mbox{ then }
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| c_q(m) =
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| c_q(n)
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| .
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| </math>
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| For a fixed value of ''q'' the absolute value of the sequence
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| :''c''<sub>''q''</sub>(1), ''c''<sub>''q''</sub>(2), ... is bounded by φ(''q''), and
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| for a fixed value of ''n'' the absolute value of the sequence
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| :''c''<sub>1</sub>(''n''), ''c''<sub>2</sub>(''n''), ... is bounded by σ(''n''), the sum of the divisors of ''n''.
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| If ''q'' > 1
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| :<math>\sum_{n=a}^{a+q-1} c_q(n)=0.
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| </math>
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| Let ''m''<sub>1</sub>, ''m''<sub>2</sub> > 0, ''m'' = lcm(''m''<sub>1</sub>, ''m''<sub>2</sub>). Then<ref>Tóth, external links, eq. 6</ref> Ramanujan's sums satisfy an [[orthogonality|orthogonality property]]:
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| :<math>
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| \frac{1}{m}\sum_{k=1}^m c_{m_1}(k) c_{m_2}(k) =
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| \begin{cases}
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| \phi(m), & \text{if }\;m_1=m_2=m,\\
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| 0, & \text{otherwise.}
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| \end{cases}
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| </math>
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| Let ''n'', ''k'' > 0. Then<ref>Tóth, external links, eq. 17.</ref>
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| :<math>
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| \sum_\stackrel{d\mid n}{\gcd(d,k)=1} d\;\frac{\mu(\tfrac{n}{d})}{\phi(d)} =
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| \frac{\mu(n) c_n(k)}{\phi(n)},
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| </math>
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| known as the [[Richard Brauer|Brauer]] - [[Hans rademacher|Rademacher]] identity.
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| If ''n'' > 0 and ''a'' is any integer, we also have<ref>Tóth, external links, eq. 8.</ref>
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| :<math>
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| \sum_\stackrel{1\le k\le n}{\gcd(k,n)=1} c_n(k-a) =
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| \mu(n)c_n(a),
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| </math>
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| due to Cohen.
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| ==Table==
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| {|
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|
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| |}
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| ==Ramanujan expansions==
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| If ''f''(''n'') is an [[arithmetic function]] (i.e. a complex-valued function of the integers or natural numbers), then a [[series (mathematics)|convergent infinite series]] of the form
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| :<math>f(n)=\sum_{q=1}^\infty a_q c_q(n)</math> or of the form
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| :<math>f(q)=\sum_{n=1}^\infty a_n c_q(n)</math> (where the ''a''<sub>''k''</sub> are complex numbers),
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| is called a '''Ramanujan expansion'''<ref>B. Berndt, commentary to ''On certain trigonometrical sums...'', Ramanujan, ''Papers'', pp. 369–371</ref> of ''f''(''n''). .
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| 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).<ref>Ramanujan, ''On certain trigonometrical sums...'' <blockquote>The majority of my formulae are "elementary" in the technical sense of the word — they can (that is to say) be proved by a combination of processes involving only finite algebra and simple general theorems concerning infinite series</blockquote>(''Papers'', p. 179)</ref><ref>The theory of formal Dirichlet series is discussed in Hardy & Wright, § 17.6 and in Knopfmacher.</ref><ref>Knopfmacher, ch. 7, discusses Ramanujan expansions as a type of Fourier expansion in an inner product space which has the ''c''<sub>''q''</sub> as an orthogonal basis.</ref>
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| The expansion of the '''zero function''' depends on a result from the analytic theory of prime numbers, namely that the series <math>\sum_{n=1}^\infty\frac{\mu(n)}{n}</math> converges to 0, and the results for ''r''(''n'') and ''r''′(''n'') depend on theorems in an earlier paper.<ref>Ramanujan, ''On Certain Arithmetical Functions''</ref>
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| All the formulas in this section are from Ramanujan's 1918 paper.
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|
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| ===Generating functions===
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| The [[generating function]]s of the Ramanujan sums are [[Dirichlet series]]:
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| :<math>
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| \zeta(s)
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| \sum_{\delta\,\mid\,q}
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| \mu\left(\frac{q}{\delta}\right)
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| \delta^{1-s} =
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| \sum_{n=1}^\infty
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| \frac{c_q(n)}{n^s}
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| </math>
| |
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| is a generating function for the sequence ''c''<sub>''q''</sub>(1), ''c''<sub>''q''</sub>(2), ... where ''q'' is kept constant, and
| |
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| :<math>
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| \frac{\sigma_{r-1}(n)}{n^{r-1}\zeta(r)}=
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| \sum_{q=1}^\infty
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| \frac{c_q(n)}{q^{r}}
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| </math>
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| is a generating function for the sequence ''c''<sub>1</sub>(''n''), ''c''<sub>2</sub>(''n''), ... where ''n'' is kept constant.
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| There is also the double Dirichlet series
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| :<math>
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| \frac{\zeta(s) \zeta(r+s-1)}{\zeta(r)}=
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| \sum_{q=1}^\infty \sum_{n=1}^\infty
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| \frac{c_q(n)}{q^r n^s}
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| .
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| </math>
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| ===σ<sub>''k''</sub>(''n'')===
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| σ<sub>''k''</sub>(''n'') is the [[divisor function]] (i.e. the sum of the ''k''<sup>th</sup> powers of the divisors of ''n'', including 1 and ''n''). σ<sub>0</sub>(n), the number of divisors of ''n'', is usually written ''d''(''n'') and σ<sub>1</sub>(n), the sum of the divisors of ''n'', is usually written σ(''n'').
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| If ''s'' > 0,
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| :<math>
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| \sigma_s(n)=
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| n^s
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| \zeta(s+1)
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| \left(
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| \frac{c_1(n)}{1^{s+1}}+
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| \frac{c_2(n)}{2^{s+1}}+
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| \frac{c_3(n)}{3^{s+1}}+
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| \dots
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| \right)
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| </math>
| |
| | |
| and
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| :<math>
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| \sigma_{-s}(n)=
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| \zeta(s+1)
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| \left(
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| \frac{c_1(n)}{1^{s+1}}+
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| \frac{c_2(n)}{2^{s+1}}+
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| \frac{c_3(n)}{3^{s+1}}+
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| \dots
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| \right).
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| </math>
| |
| | |
| Setting ''s'' = 1 gives
| |
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| :<math>
| |
| \sigma(n)=
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| \frac{\pi^2}{6}n
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| \left(
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| \frac{c_1(n)}{1}+
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| \frac{c_2(n)}{4}+
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| \frac{c_3(n)}{9}+
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| \dots
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| \right) .
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| </math>
| |
| | |
| If the [[Riemann hypothesis]] is true, and <math>-\tfrac12<s<\tfrac12,</math>
| |
| | |
| :<math>
| |
| \begin{align}
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| \sigma_s(n)
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| &=
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| \zeta(1-s)
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| \left(
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| \frac{c_1(n)}{1^{1-s}}+
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| \frac{c_2(n)}{2^{1-s}}+
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| \frac{c_3(n)}{3^{1-s}}+
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| \dots
| |
| \right)\\
| |
| | |
| &=
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| n^s
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| \zeta(1+s)
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| \left(
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| \frac{c_1(n)}{1^{1+s}}+
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| \frac{c_2(n)}{2^{1+s}}+
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| \frac{c_3(n)}{3^{1+s}}+
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| \dots
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| \right).\\
| |
| \end{align}
| |
| </math>
| |
| | |
| ===''d''(''n'')===
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| ''d''(''n'') = σ<sub>0</sub>(''n'') is the number of divisors of ''n'', including 1 and ''n'' itself.
| |
| | |
| :<math>
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| -d(n)=
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| \frac{\log 1}{1}c_1(n)+
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| \frac{\log 2}{2}c_2(n)+
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| \frac{\log 3}{3}c_3(n)+
| |
| \dots
| |
| </math>
| |
| | |
| and
| |
| | |
| :<math>
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| -d(n)(2\gamma+\log n)=
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| \frac{\log^2 1}{1}c_1(n)+
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| \frac{\log^2 2}{2}c_2(n)+
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| \frac{\log^2 3}{3}c_3(n)+
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| \dots
| |
| </math>
| |
| | |
| where γ = 0.5772... is the [[Euler–Mascheroni constant]].
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| ===φ(''n'')===
| |
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| [[Euler's totient function]] φ(''n'') is the number of positive integers less than ''n'' and coprime to ''n''.
| |
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| Ramanujan defines a generalization of it: if <math>n=p_1^{a_1}p_2^{a_2}p_3^{a_3}\dots</math> is the prime factorization of ''n'', and ''s'' is a complex number, let
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| :<math>\phi_s(n)=n^s(1-p_1^{-s})(1-p_2^{-s})(1-p_3^{-s})\dots,
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| </math> so that φ<sub>1</sub>(''n'') = φ(''n'') is Euler's function.<ref>This is [[Jordan's totient function]], J<sub>''s''</sub>(''n'').</ref>
| |
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| He proves that
| |
| | |
| :<math>
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| \frac{\mu(n)n^s}{\phi_s(n)\zeta(s)}=
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| \sum_{\nu=1}^\infty \frac{\mu(n\nu)}{\nu^s}
| |
| </math>
| |
| | |
| and uses this to show that
| |
| | |
| :<math>\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)}+\dots.
| |
| </math>
| |
| | |
| Letting ''s'' = 1,
| |
| | |
| :<math>
| |
| | |
| \begin{align}
| |
| | |
| \phi(n) =
| |
| | |
| \frac{6}{\pi^2}n
| |
| | |
| \Big(
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| c_1(n)
| |
| | |
| &-\frac{c_2(n)}{2^2-1}
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| -\frac{c_3(n)}{3^2-1}
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| -\frac{c_5(n)}{5^2-1} \\
| |
| | |
| &+\frac{c_6(n)}{(2^2-1)(3^2-1)}
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| -\frac{c_7(n)}{7^2-1}
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| +\frac{c_{10}(n)}{(2^2-1)(5^2-1)}
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| -\dots
| |
| \Big).\\
| |
| \end{align}
| |
| </math>
| |
| | |
| Note that the constant is the inverse<ref>Cf. Hardy & Wright, Thm. 329, which states that
| |
| <math>\;\frac{6}{\pi^2}<\frac{\sigma(n)\phi(n)}{n^2}<1.</math></ref> of the one in the formula for σ(''n'').
| |
| | |
| ===Λ(''n'')===
| |
| | |
| [[von Mangoldt function|Von Mangoldt's function]] Λ(n) is zero unless ''n'' = ''p''<sup>''k''</sup> is a power of a prime number, in which case it is the natural logarithm log ''p''.
| |
| | |
| :<math>
| |
| -\Lambda(m) =
| |
| c_m(1)+
| |
| \frac12c_m(2)+
| |
| \frac13c_m(3)+
| |
| \dots
| |
| </math>
| |
| | |
| ===Zero===
| |
| For all ''n'' > 0,
| |
| | |
| :<math>0=
| |
| c_1(n)+
| |
| \frac12c_2(n)+
| |
| \frac13c_3(n)+
| |
| \dots.
| |
| </math>
| |
| | |
| This is equivalent to the [[prime number theorem]].<ref>Hardy, ''Ramanujan'', p. 141</ref><ref>B. Berndt, commentary to ''On certain trigonometrical sums...'', Ramanujan, ''Papers'', p. 371</ref>
| |
| | |
| ===''r''<sub>2''s''</sub>(''n'') (sums of squares)===
| |
| ''r''<sub>2''s''</sub>(''n'') is the number of way of representing ''n'' as the sum of 2''s'' [[Square number|squares]], counting different orders and signs as different (e.g., ''r''<sub>2</sub>(13) = 8, as 13 = (±2)<sup>2</sup> + (±3)<sup>2</sup> = (±3)<sup>2</sup> + (±2)<sup>2</sup>.)
| |
| | |
| Ramanujan defines a function δ<sub>2''s''</sub>(''n'') and references a paper<ref>Ramanujan, ''On Certain Arithmetical Functions''</ref> in which he proved that ''r''<sub>2''s''</sub>(''n'') = δ<sub>2''s''</sub>(''n'') for ''s'' = 1, 2, 3, and 4. For ''s'' > 4 he shows that δ<sub>2''s''</sub>(''n'') is a good approximation to ''r''<sub>2''s''</sub>(''n'').
| |
| | |
| ''s'' = 1 has a special formula:
| |
| | |
| :<math>
| |
| \delta_2(n)=
| |
| \pi
| |
| \left(
| |
| \frac{c_1(n)}{1}-
| |
| \frac{c_3(n)}{3}+
| |
| \frac{c_5(n)}{5}-
| |
| \dots
| |
| \right).
| |
| </math>
| |
| | |
| In the following formulas the signs repeat with a period of 4.
| |
| | |
| If ''s'' ≡ 0 (mod 4),
| |
| :<math>
| |
| \delta_{2s}(n)=
| |
| \frac{\pi^s n^{s-1}}{(s-1)!}
| |
| \left(
| |
| \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}+
| |
| \dots
| |
| \right)
| |
| </math>
| |
| | |
| If ''s'' ≡ 2 (mod 4),
| |
| :<math>
| |
| \delta_{2s}(n)=
| |
| \frac{\pi^s n^{s-1}}{(s-1)!}
| |
| \left(
| |
| \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}+
| |
| \dots
| |
| \right)
| |
| </math>
| |
| | |
| If ''s'' ≡ 1 (mod 4) and ''s'' > 1,
| |
| :<math>
| |
| \delta_{2s}(n)=
| |
| \frac{\pi^s n^{s-1}}{(s-1)!}
| |
| \left(
| |
| \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}+
| |
| \dots
| |
| \right)
| |
| </math>
| |
| | |
| If ''s'' ≡ 3 (mod 4),
| |
| :<math>
| |
| \delta_{2s}(n)=
| |
| \frac{\pi^s n^{s-1}}{(s-1)!}
| |
| \left(
| |
| \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}+
| |
| \dots
| |
| \right)
| |
| </math>
| |
| | |
| and therefore,
| |
| | |
| :<math>
| |
| r_2(n)=
| |
| \pi
| |
| \left(
| |
| \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}+
| |
| \dots
| |
| \right)
| |
| </math>
| |
| | |
| :<math>
| |
| r_4 (n)=
| |
| \pi^2 n
| |
| \left(
| |
| \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}+
| |
| \dots
| |
| \right)
| |
| </math>
| |
| | |
| :<math>
| |
| r_6(n)=
| |
| \frac{\pi^3 n^2}{2}
| |
| \left(
| |
| \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}+
| |
| \dots
| |
| \right)
| |
| </math>
| |
| | |
| :<math>
| |
| r_8(n)=
| |
| \frac{\pi^4 n^3}{6}
| |
| \left(
| |
| \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}+
| |
| \dots
| |
| \right)
| |
| </math>
| |
| | |
| ===''r''′<sub>2''s''</sub>(''n'') (sums of triangles)===
| |
| ''r''′<sub>2''s''</sub>(''n'') is the number of ways ''n'' can be represented as the sum of 2''s'' [[triangular number]]s (i.e. the numbers 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, 15, ...; the ''n''<sup>th</sup> 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′<sub>2''s''</sub>(''n'') such that ''r''′<sub>2''s''</sub>(''n'') = δ′<sub>2''s''</sub>(''n'') for ''s'' = 1, 2, 3, and 4, and that for ''s'' > 4, δ′<sub>2''s''</sub>(''n'') is a good approximation to ''r''′<sub>2''s''</sub>(''n'').
| |
| | |
| Again, ''s'' = 1 requires a special formula:
| |
| | |
| :<math>
| |
| \delta'_2(n)=
| |
| \frac{\pi}{4}
| |
| \left(
| |
| \frac{c_1(4n+1)}{1}-
| |
| \frac{c_3(4n+1)}{3}+
| |
| \frac{c_5(4n+1)}{5}-
| |
| \frac{c_7(4n+1)}{7}+
| |
| \dots
| |
| \right).
| |
| </math>
| |
| | |
| If ''s'' is a multiple of 4,
| |
| :<math>
| |
| \delta'_{2s}(n)=
| |
| \frac{(\frac12\pi)^s}{(s-1)!}\left(n+\frac{s}4\right)^{s-1}
| |
| \left(
| |
| \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}+
| |
| \dots
| |
| \right).
| |
| </math>
| |
| | |
| If ''s'' is twice an odd number,
| |
| :<math>
| |
| \delta'_{2s}(n)=
| |
| \frac{(\frac12\pi)^s}{(s-1)!}\left(n+\frac{s}4\right)^{s-1}
| |
| \left(
| |
| \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}+
| |
| \dots
| |
| \right).
| |
| </math>
| |
| | |
| If ''s'' is an odd number and ''s'' > 1,
| |
| :<math>
| |
| \delta'_{2s}(n)=
| |
| \frac{(\frac12\pi)^s}{(s-1)!}\left(n+\frac{s}4\right)^{s-1}
| |
| \left(
| |
| \frac{c_1(4n+s)}{1^s}-
| |
| \frac{c_3(4n+s)}{3^s}+
| |
| \frac{c_5(4n+s)}{5^s}-
| |
| \dots
| |
| \right).
| |
| </math>
| |
| | |
| Therefore,
| |
| | |
| :<math> | |
| r'_2(n)=
| |
| \frac{\pi}{4}
| |
| \left(
| |
| \frac{c_1(4n+1)}{1}-
| |
| \frac{c_3(4n+1)}{3}+
| |
| \frac{c_5(4n+1)}{5}-
| |
| \frac{c_7(4n+1)}{7}+
| |
| \dots
| |
| \right)
| |
| </math>
| |
| | |
| :<math>
| |
| r'_4(n)=
| |
| \left(\tfrac12\pi\right)^2\left(n+\tfrac12\right)
| |
| \left(
| |
| \frac{c_1(2n+1)}{1}+
| |
| \frac{c_3(2n+1)}{9}+
| |
| \frac{c_5(2n+1)}{25}+
| |
| \dots
| |
| \right)
| |
| </math>
| |
| | |
| :<math>
| |
| r'_6(n)=
| |
| \frac{(\frac12\pi)^3}{2}\left(n+\tfrac34\right)^2
| |
| \left(
| |
| \frac{c_1(4n+3)}{1}-
| |
| \frac{c_3(4n+3)}{27}+
| |
| \frac{c_5(4n+3)}{125}-
| |
| \dots
| |
| \right)
| |
| </math>
| |
| | |
| :<math>
| |
| r'_8(n)=
| |
| \frac{(\frac12\pi)^4}{6}(n+1)^3
| |
| \left(
| |
| \frac{c_1(n+1)}{1}+
| |
| \frac{c_3(n+1)}{81}+
| |
| \frac{c_5(n+1)}{625}+
| |
| \dots
| |
| \right).
| |
| </math>
| |
| | |
| ===Sums===
| |
| | |
| Let
| |
| | |
| :<math>
| |
| T_q(n) =
| |
| c_q(1) +
| |
| c_q(2)+
| |
| \dots+c_q(n)
| |
| </math>
| |
| | |
| and
| |
| | |
| :<math>
| |
| U_q(n) =
| |
| T_q(n) +
| |
| \tfrac12\phi(q).
| |
| </math>
| |
| | |
| Then if ''s'' > 1,
| |
| :<math>
| |
| \sigma_{-s}(1)+
| |
| \sigma_{-s}(2)+
| |
| \dots+
| |
| \sigma_{-s}(n)
| |
| </math>
| |
| | |
| ::<math>=
| |
| \zeta(s+1)
| |
| \left(
| |
| n+
| |
| \frac{T_2(n)}{2^{s+1}}+
| |
| \frac{T_3(n)}{3^{s+1}}+
| |
| \frac{T_4(n)}{4^{s+1}}
| |
| +\dots
| |
| \right)
| |
| </math>
| |
| | |
| ::<math>=
| |
| \zeta(s+1)
| |
| \left(
| |
| n+\tfrac12+
| |
| \frac{U_2(n)}{2^{s+1}}+
| |
| \frac{U_3(n)}{3^{s+1}}+
| |
| \frac{U_4(n)}{4^{s+1}}
| |
| +\dots
| |
| \right)-
| |
| \tfrac12\zeta(s)
| |
| ,
| |
| </math>
| |
| | |
| :<math>
| |
| d(1)+
| |
| d(2)+
| |
| \dots+
| |
| d(n)
| |
| </math>
| |
| | |
| ::<math>=
| |
| -\frac{T_2(n)\log2}{2}
| |
| -\frac{T_3(n)\log3}{3}
| |
| -\frac{T_4(n)\log4}{4}
| |
| -\dots
| |
| ,
| |
| </math>
| |
| | |
| :<math>
| |
| d(1)\log1+
| |
| d(2)\log2+
| |
| \dots+
| |
| d(n)\log n
| |
| </math>
| |
| | |
| ::<math>=
| |
| -\frac{T_2(n)(2\gamma\log2-\log^22)}{2}
| |
| -\frac{T_3(n)(2\gamma\log3-\log^23)}{3}
| |
| -\frac{T_4(n)(2\gamma\log4-\log^24)}{4}
| |
| -\dots
| |
| ,
| |
| </math>
| |
| | |
| :<math>
| |
| r_2(1)+
| |
| r_2(2)+
| |
| \dots+
| |
| r_2(n)
| |
| </math>
| |
| | |
| ::<math>=
| |
| \pi
| |
| \left(
| |
| n
| |
| -\frac{T_3(n)}{3}
| |
| +\frac{T_5(n)}{5}
| |
| -\frac{T_7(n)}{7}
| |
| +\dots
| |
| \right)
| |
| .
| |
| </math>
| |
| | |
| ==See also==
| |
| | |
| *[[Gaussian period]]
| |
| *[[Kloosterman sum]]
| |
| | |
| ==Notes==
| |
| | |
| {{reflist}}
| |
| | |
| ==References==
| |
| | |
| *{{citation
| |
| | last1 = Hardy | first1 = G. H.
| |
| | title = Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work
| |
| | publisher = AMS / Chelsea
| |
| | location = Providence RI
| |
| | year = 1999
| |
| | isbn = 978-0-8218-2023-0}}
| |
| | |
| *{{citation
| |
| | last1 = Hardy | first1 = G. H.
| |
| | last2 = Wright | first2 = E. M.
| |
| | title = An Introduction to the Theory of Numbers (Fifth edition)
| |
| | publisher = [[Oxford University Press]]
| |
| | location = Oxford
| |
| | year = 1980
| |
| | isbn = 978-0-19-853171-5}}
| |
| | |
| *{{citation
| |
| | last1 = Knopfmacher | first1 = John
| |
| | title = Abstract Analytic Number Theory
| |
| | publisher = Dover
| |
| | location = New York
| |
| | year = 1990
| |
| | isbn = 0-486-66344-2}}
| |
| | |
| *{{citation
| |
| | title=Additive Number Theory: the Classical Bases
| |
| | volume=164
| |
| | series=Graduate Texts in Mathematics
| |
| | last=Nathanson
| |
| | first=Melvyn B.
| |
| | publisher=Springer-Verlag
| |
| | year=1996 | isbn=0-387-94656-X }} Section A.7.
| |
| | |
| *{{citation
| |
| | last1 = Ramanujan | first1 = Srinivasa
| |
| | title = On Certain Trigonometric Sums and their Applications in the Theory of Numbers
| |
| | journal = Transactions of the Cambridge Philosophical Society
| |
| | volume = 22
| |
| | issue = 15
| |
| | year = 1918
| |
| | pages = 259–276}} (pp. 179–199 of his ''Collected Papers'')
| |
| | |
| *{{citation
| |
| | last1 = Ramanujan | first1 = Srinivasa
| |
| | title = On Certain Arithmetical Functions
| |
| | journal = Transactions of the Cambridge Philosophical Society
| |
| | volume = 22
| |
| | issue = 9
| |
| | year = 1916
| |
| | pages = 159–184}} (pp. 136–163 of his ''Collected Papers'')
| |
| | |
| *{{citation
| |
| | last1 = Ramanujan | first1 = Srinivasa
| |
| | title = Collected Papers
| |
| | publisher = AMS / Chelsea
| |
| | location = Providence RI
| |
| | year = 2000
| |
| | isbn = 978-0-8218-2076-6}}
| |
| | |
| ==External links==
| |
| * László Tóth, [http://arxiv.org/pdf/1104.1906.pdf Sums of products of Ramanujan sums]
| |
| | |
| [[Category:Number theory]]
| |
| [[Category:Srinivasa Ramanujan]]
| |