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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.〔Ramanujan, ''On Certain Trigonometric Sums ...'' 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.(''Papers'', p. 179). In a footnote cites pp. 360–370 of the Dirichlet-Dedekind ''Vorlesungen über Zahlentheorie'', 4th ed.〕 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.〔Nathanson, ch. 8〕 ==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 divisor, is Euler's totient function, is the Möbius function, and is the Riemann zeta function. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ramanujan's sum」の詳細全文を読む スポンサード リンク
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