# Ramanujan sum

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##### 1: 27.10 Periodic Number-Theoretic Functions
An example is Ramanujan’s sum:
27.10.5 $c_{k}\left(n\right)=\sum_{d\mathbin{|}\left(n,k\right)}d\mu\left(\frac{k}{d}% \right).$
27.10.8 $a_{k}(m)=\sum_{d\mathbin{|}\left(m,k\right)}g(d)f\left(\frac{k}{d}\right)\frac% {d}{k}.$
In particular, $G\left(n,\chi_{1}\right)=c_{k}\left(n\right)$. …
##### 2: Bibliography B
• A. Basu and T. M. Apostol (2000) A new method for investigating Euler sums. Ramanujan J. 4 (4), pp. 397–419.
• ##### 3: 27.14 Unrestricted Partitions
27.14.18 $x\prod_{n=1}^{\infty}(1-x^{n})^{24}=\sum_{n=1}^{\infty}\tau\left(n\right)x^{n},$ $|x|<1$.
27.14.19 $\tau\left(m\right)\tau\left(n\right)=\sum_{d\mathbin{|}\left(m,n\right)}d^{11}% \tau\left(\frac{mn}{d^{2}}\right),$ $m,n=1,2,\dots$.
##### 4: 20.11 Generalizations and Analogs
###### §20.11(ii) Ramanujan’s Theta Function and $q$-Series
Ramanujan’s theta function $f(a,b)$ is defined by …
###### §20.11(iii) Ramanujan’s Change of Base
These results are called Ramanujan’s changes of base. …
##### 5: Bibliography M
• S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.
• S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.
• ##### 6: 20.12 Mathematical Applications
For applications of $\theta_{3}\left(0,q\right)$ to problems involving sums of squares of integers see §27.13(iv), and for extensions see Estermann (1959), Serre (1973, pp. 106–109), Koblitz (1993, pp. 176–177), and McKean and Moll (1999, pp. 142–143). For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s $\tau\left(n\right)$ function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). …
##### 7: 17.2 Calculus
17.2.45 $\int_{0}^{1}f(x){\mathrm{d}}_{q}x=(1-q)\sum_{j=0}^{\infty}f(q^{j})q^{j},$
17.2.46 $\int_{0}^{a}f(x){\mathrm{d}}_{q}x=a(1-q)\sum_{j=0}^{\infty}f(aq^{j})q^{j}.$
provided that $\sum_{j=-\infty}^{\infty}f(q^{j})q^{j}$ converges.
##### 8: 26.10 Integer Partitions: Other Restrictions
where the last right-hand side is the sum over $m\geq 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … where the inner sum is the sum of all positive odd divisors of $t$. … where the inner sum is the sum of all positive divisors of $t$ that are in $S$.
###### §26.10(iv) Identities
Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. …
##### 9: 17.12 Bailey Pairs
17.12.1 $\sum_{n=0}^{\infty}\alpha_{n}\gamma_{n}=\sum_{n=0}^{\infty}\beta_{n}\delta_{n},$
$\beta_{n}=\sum_{j=0}^{n}\alpha_{j}u_{n-j}v_{n+j},$
$\gamma_{n}=\sum_{j=n}^{\infty}\delta_{j}u_{j-n}v_{j+n}.$
The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: …
##### 10: Bibliography L
• D. H. Lehmer (1943) Ramanujan’s function $\tau(n)$ . Duke Math. J. 10 (3), pp. 483–492.
• D. H. Lehmer (1947) The vanishing of Ramanujan’s function $\tau(n)$ . Duke Math. J. 14 (2), pp. 429–433.
• J. Lepowsky and R. L. Wilson (1982) A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. in Math. 45 (1), pp. 21–72.
• M. Lerch (1887) Note sur la fonction $\mathfrak{K}(w,x,s)=\sum_{k=0}^{\infty}\frac{e^{2k\pi ix}}{(w+k)^{s}}$ . Acta Math. 11 (1-4), pp. 19–24 (French).