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! ''n'' !! prime factorization !! {{sigma}}<sub>0</sub>(''n'')!! {{sigma}}<sub>1</sub>(''n'')!! {{sigma}}<sub>2</sub>(''n'')!! {{sigma}}<sub>3</sub>(''n'')!! {{sigma}}<sub>4</sub>(''n'')
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is true for infinitely many values of {{mvar|n}}, see {{OEIS2C|A005237}}.
=== Dirichlet convolutions ===
{{Main article|Dirichlet convolution}}
By definition:<math display="block">\sigma = \operatorname{Id} * \mathbf 1</math>By [[Möbius inversion formula|Möbius inversion]]:<math display="block">\operatorname{Id} = \sigma * \mu </math>
==Series relations==
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A [[Lambert series]] involving the divisor function is: {{sfnp|Hardy|Wright|2008|pp=338-341|loc=§17.10}}
:<math>\sum_{n=1}^\infty q^n \sigma_a(n) = \sum_{n=1}^\infty \sum_{j=1}^\infty n^a q^{j\,n} = \sum_{n=1}^\infty \frac{n^a q^n}{1-q^n} = \sum_{n=1}^\infty \operatorname{Li}_{-a}(q^n)</math>
for arbitrary [[complex number|complex]] |''q''| ≤ 1 and ''a'' (<math>\operatorname{Li}</math> is the [[polylogarithm]]). This summation also appears as the [[Eisenstein series#Fourier series|Fourier series of the Eisenstein series]] and the [[Weierstrass elliptic functions#Invariants|invariants of the Weierstrass elliptic functions]].
For <math>k>0</math>, there is an explicit series representation with [[Ramanujan sum]]s <math> c_m(n) </math> as :<ref>{{cite book |author=E. Krätzel |title=Zahlentheorie |publisher=VEB Deutscher Verlag der Wissenschaften |place =Berlin |year=1981 |pages=130}} (German)</ref>
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| year = 1967| issue = 8
}}
* {{Citation | last1=Grönwall | first1=Thomas Hakon | author1-link=Thomas Hakon Grönwall | title=Some asymptotic expressions in the theory of numbers | year=1913 | journal=Transactions of the American Mathematical Society | volume=14 | issue=1 | pages=113–122 | doi=10.1090/S0002-9947-1913-1500940-6| doi-access=free }}
* {{Citation | last1=Hardy | first1=G. H. | author1-link=G. H. Hardy | last2=Wright | first2=E. M. | author2-link=E. M. Wright | edition=6th | others=Revised by [[Roger Heath-Brown|D. R. Heath-Brown]] and [[Joseph H. Silverman|J. H. Silverman]]. Foreword by [[Andrew Wiles]]. | title=An Introduction to the Theory of Numbers | publisher=[[Oxford University Press]] | ___location=Oxford | isbn=978-0-19-921986-5 | mr=2445243 | zbl=1159.11001 | year=2008 | orig-year=1938}}
* {{citation | last=Ivić | first=Aleksandar | title=The Riemann zeta-function. The theory of the Riemann zeta-function with applications | series=A Wiley-Interscience Publication | ___location=New York etc. | publisher=John Wiley & Sons | year=1985 | isbn=0-471-80634-X | zbl=0556.10026 | pages=385–440 }}
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