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| {{distinguish|Eisenstein sum}}
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| :''This article describes '''holomorphic''' Eisenstein series; for the non-holomorphic case see [[real analytic Eisenstein series]]''
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| '''Eisenstein series''', named after [[Germany|German]] [[mathematician]] [[Gotthold Eisenstein]], are particular [[modular form]]s with [[infinite series]] expansions that may be written down directly. Originally defined for the [[modular group]], Eisenstein series can be generalized in the theory of [[automorphic form]]s.
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| ==Eisenstein series for the modular group==
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| [[Image:Gee_three_real.jpeg|thumb|The real part of ''G''<sub>6</sub> as a function of ''q'' on the [[unit disk]].]] | |
| [[Image:Gee_three_imag.jpeg|thumb|The imaginary part of ''G''<sub>6</sub> as a function of ''q'' on the unit disk.]]
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| Let τ be a [[complex number]] with strictly positive [[imaginary part]]. Define the '''holomorphic Eisenstein series''' ''G''<sub>2''k''</sub>(τ) of weight 2''k'', where ''k'' ≥ 2 is an integer, by the following series:
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| :<math>G_{2k}(\tau) = \sum_{ (m,n)\in\mathbf{Z}^2\backslash(0,0)} \frac{1}{(m+n\tau )^{2k}}.</math>
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| This series [[absolute convergence|absolutely converges]] to a holomorphic function of τ in the [[upper half-plane]] and its Fourier expansion given below shows that it extends to a holomorphic function at τ = i∞. It is a remarkable fact that the Eisenstein series is a [[modular form]]. Indeed, the key property is its SL(2, '''Z''')-invariance. Explicitly if ''a'', ''b'', ''c'', ''d'' ∈ '''Z''' and ''ad''−''bc'' = 1 then
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| :<math>G_{2k} \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2k} G_{2k}(\tau)</math>
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| and ''G''<sub>2''k''</sub> is therefore a modular form of weight 2''k''. Note that it is important to assume that ''k'' ≥ 2, otherwise it would be illegitimate to change the order of summation, and the SL(2, '''Z''')-invariance would not hold. In fact, there are no nontrivial modular forms of weight 2. Nevertheless, an analogue of the holomorphic Eisenstein series can be defined even for ''k'' = 1, although it would only be a [[quasimodular form]]. | |
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| ==Relation to modular invariants==
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| The [[Elliptic modular function|modular invariants]] ''g''<sub>2</sub> and ''g''<sub>3</sub> of an [[elliptic curve]] are given by the first two terms of the Eisenstein series as
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| :<math>g_2 = 60 G_4</math> | |
| :<math>g_3 = 140 G_6</math>
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| The article on modular invariants provides expressions for these two functions in terms of [[theta function]]s.
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| ==Recurrence relation==
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| Any holomorphic modular form for the modular group can be written as a polynomial in ''G''<sub>4</sub> and ''G''<sub>6</sub>. Specifically, the higher order ''G''<sub>2''k''</sub>'s can be written in terms of ''G''<sub>4</sub> and ''G''<sub>6</sub> through a recurrence relation. Let ''d<sub>k</sub>'' =(2''k''+3)''k''!''G''<sub>2''k''+4</sub>. Then the ''d<sub>k</sub>'' satisfy the relation
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| :<math>\sum_{k=0}^n {n \choose k} d_k d_{n-k} = \frac{2n+9}{3n+6}d_{n+2}</math>
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| for all ''n'' ≥ 0. Here, <math>{n \choose k}</math> is the [[binomial coefficient]] and <math>d_0=3G_4</math> and <math>d_1=5G_6</math>.
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| The ''d''<sub>''k''</sub> occur in the series expansion for the [[Weierstrass's elliptic functions]]:
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| :<math>\wp(z) =\frac{1}{z^2} + z^2 \sum_{k=0}^\infty \frac {d_k z^{2k}}{k!} =\frac{1}{z^2} + \sum_{k=1}^\infty (2k+1) G_{2k+2} z^{2k}.</math>
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| ==Fourier series==
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| [[Image:Eisenstein_4.jpg|right|thumb|G_4]]
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| [[Image:Eisenstein_6.jpg|right|thumb|G_6]]
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| [[Image:Eisenstein_8.jpg|right|thumb|G_8]]
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| [[Image:Eisenstein_10.jpg|right|thumb|G_10]]
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| [[Image:Eisenstein_12.jpg|right|thumb|G_12]]
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| [[Image:Eisenstein_14.jpg|right|thumb|G_14]]
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| Define <math>q=e^{2\pi i\tau}</math>. (Some older books define ''q'' to be the [[nome (mathematics)|nome]] <math>q=e^{i\pi\tau}</math>, but <math>q=e^{2\pi i\tau}</math> is now standard in number theory.) Then the [[Fourier series]] of the Eisenstein series is
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| :<math>G_{2k}(\tau) = 2\zeta(2k) \left(1+c_{2k}\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{n} \right)</math>
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| where the [[Fourier coefficient]]s ''c''<sub>2''k''</sub> are given by
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| :<math>c_{2k} = \frac{(2\pi i)^{2k}}{(2k-1)! \zeta(2k)} = \frac {-4k}{B_{2k}} = \frac {2}{\zeta(1-2k)}.</math>
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| Here, ''B''<sub>''n''</sub> are the [[Bernoulli number]]s, ζ(''z'') is [[Riemann's zeta function]] and σ<sub>''p''</sub>(''n'') is the [[divisor function|divisor sum function]], the sum of the ''p''<sup>th</sup> powers of the divisors of ''n''. In particular, one has
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| :<math>\begin{align}
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| G_4(\tau)&=\frac{\pi^4}{45} \left[ 1+ 240\sum_{n=1}^\infty \sigma_3(n) q^{n} \right] \\
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| G_6(\tau)&=\frac{2\pi^6}{945} \left[ 1- 504\sum_{n=1}^\infty \sigma_5(n) q^{n} \right].
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| \end{align}</math>
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| The summation over ''q'' can be resummed as a [[Lambert series]]; that is, one has
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| :<math>\sum_{n=1}^{\infty} q^n \sigma_a(n) = \sum_{n=1}^{\infty} \frac{n^a q^n}{1-q^n}</math>
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| for arbitrary [[complex number|complex]] |''q''| ≤ 1 and ''a''. When working with the [[q-expansion]] of the Eisenstein series, the alternate notation
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| :<math>E_{2k}(\tau)=\frac{G_{2k}(\tau)}{2\zeta (2k)}= 1+\frac {2}{\zeta(1-2k)}\sum_{n=1}^{\infty} \frac{n^{2k-1} q^n}{1-q^n} = 1 - \frac{4k}{B_{2k}} \sum_{d,n \geq 1} n^{2k-1} q^{n d} </math>
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| is frequently introduced.
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| == Identities involving Eisenstein series ==
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| === Products of Eisenstein series ===
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| Eisenstein series form the most explicit examples of [[modular form]]s for the full modular group SL(2, '''Z'''). Since the space of modular forms of weight 2''k'' has dimension 1 for 2''k'' = 4, 6, 8, 10, 14, different products of Eisenstein series having those weights have to be proportional. Thus we obtain the identities:
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| :<math>E_4^2 = E_8, \quad E_4 E_6 = E_{10}, \quad E_4 E_{10} = E_{14}, \quad E_6 E_8 = E_{14}. </math>
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| Using the ''q''-expansions of the Eisenstein series given above, they may be restated as identities involving the sums of powers of divisors:
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| :<math>(1+240\sum_{n=1}^\infty \sigma_3(n) q^n)^2 = 1+480\sum_{n=1}^\infty \sigma_7(n) q^n,</math>
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| hence
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| :<math>\sigma_7(n)=\sigma_3(n)+120\sum_{m=1}^{n-1}\sigma_3(m)\sigma_3(n-m),</math>
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| and similarly for the others. Perhaps, even more interestingly, the [[theta function]] of an eight-dimensional even unimodular lattice Γ is a modular form of weight 4 for the full modular group, which gives the following identities:
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| :<math> \theta_{\Gamma}(\tau)=1+\sum_{n=1}^\infty r_{\Gamma}(2n) q^{n} = E_4(\tau), \quad r_{\Gamma}(n) = 240\sigma_3(n) </math>
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| for the number ''r''<sub>Γ</sub>(''n'') of vectors of the squared length ''2n'' in the [[E8 lattice|root lattice of the type E<sub>8</sub>]].
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| Similar techniques involving holomorphic Eisenstein series twisted by a [[Dirichlet character]] produce formulas for the number of representations of a positive integer ''n'' as a sum of two, four, or eight squares in terms of the divisors of ''n''.
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| Using the above recurrence relation, all higher ''E''<sub>2''k''</sub> can be expressed as polynomials in ''E''<sub>4</sub> and ''E''<sub>6</sub>. For example:
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| :<math>\begin{align}
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| E_{8} &= E_4^2 \\
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| E_{10} &= E_4\cdot E_6 \\
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| 691 \cdot E_{12} &= 441\cdot E_4^3+ 250\cdot E_6^2 \\
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| E_{14} &= E_4^2\cdot E_6 \\
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| 3617\cdot E_{16} &= 1617\cdot E_4^4+ 2000\cdot E_4 \cdot E_6^2 \\
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| 43867 \cdot E_{18} &= 38367\cdot E_4^3\cdot E_6+5500\cdot E_6^3 \\
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| 174611 \cdot E_{20} &= 53361\cdot E_4^5+ 121250\cdot E_4^2\cdot E_6^2 \\
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| 77683 \cdot E_{22} &= 57183\cdot E_4^4\cdot E_6+20500\cdot E_4\cdot E_6^3 \\
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| 236364091 \cdot E_{24} &= 49679091\cdot E_4^6+ 176400000\cdot E_4^3\cdot E_6^2 + 10285000\cdot E_6^4
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| \end{align}</math>
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| Many relationships between products of Eisenstein series can be written in an elegant way using [[Hankel matrix|Hankel determinants]], e.g. Garvan's identity
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| : <math> \Delta^2=-\frac{691}{1728^2\cdot250}\det \begin{vmatrix}E_4&E_6&E_8\\ E_6&E_8&E_{10}\\ E_8&E_{10}&E_{12}\end{vmatrix}</math>
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| where
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| :<math> \Delta=\frac{E_4^3-E_6^2}{1728}</math>
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| is the [[modular discriminant]]. See
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| * Steven C. Milne (2000), ''[http://arxiv.org/pdf/math/0009130v3.pdf Hankel Determinants of Eisenstein Series]'', Ohio State University.
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| === Ramanujan identities ===
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| [[Ramanujan]] gave several interesting identities between the first few Eisenstein series involving differentiation. Let
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| :<math>L(q)=1-24\sum_{n=1}^\infty \frac {nq^n}{1-q^n}=E_2(\tau)</math>
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| :<math>M(q)=1+240\sum_{n=1}^\infty \frac {n^3q^n}{1-q^n}=E_4(\tau)</math>
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| :<math>N(q)=1-504\sum_{n=1}^\infty \frac {n^5q^n}{1-q^n}=E_6(\tau),</math>
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| then
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| :<math>q\frac{dL}{dq} = \frac {L^2-M}{12}</math>
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| :<math>q\frac{dM}{dq} = \frac {LM-N}{3}</math>
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| :<math>q\frac{dN}{dq} = \frac {LN-M^2}{2}.</math>
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| These identities, like the identities between the series, yield arithmetical [[convolution]] identities involving the [[divisor function|sum-of-divisor function]]. Following Ramanujan, to put these identities in the simplest form it is necessary to extend the domain of σ<sub>''p''</sub>(''n'') to include zero, by setting
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| :<math>\sigma_p(0) = \frac12\zeta(-p).\;</math> E.g.:
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| ::<math>\begin{align}
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| \sigma(0) &= -\frac{1}{24}\\
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| \sigma_3(0) &= \frac{1}{240}\\
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| \sigma_5(0) &= -\frac{1}{504}.
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| \end{align}</math>
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| Then, for example
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| :<math>\sum_{k=0}^n\sigma(k)\sigma(n-k)=\frac5{12}\sigma_3(n)-\frac12n\sigma(n).</math>
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| Other identities of this type, but not directly related to the preceding relations between ''L'', ''M'' and ''N'' functions, have been proved by [[Ramanujan]] and [[Giuseppe Melfi|Melfi]], as for example
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| :<math> \sum_{k=0}^n\sigma_3(k)\sigma_3(n-k)=\frac1{120}\sigma_7(n)</math>
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| :<math> \sum_{k=0}^n\sigma(2k+1)\sigma_3(n-k)=\frac1{240}\sigma_5(2n+1)</math>
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| :<math> \sum_{k=0}^n\sigma(3k+1)\sigma(3n-3k+1)=\frac19\sigma_3(3n+2).</math>
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| For a comprehensive list of convolution identities involving sum-of-divisors functions and related topics see
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| * [[S. Ramanujan]], ''On certain arithmetical functions'', pp 136-162, reprinted in ''Collected Papers'', (1962), Chelsea, New York.
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| * Heng Huat Chan and Yau Lin Ong, ''[http://www.ams.org/proc/1999-127-06/S0002-9939-99-04832-7/S0002-9939-99-04832-7.pdf On Eisenstein Series]'', (1999) Proceedings of the Amer. Math. Soc. '''127'''(6) pp.1735-1744
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| * [[Giuseppe Melfi|G. Melfi]], ''On some modular identities'', in Number Theory, Diophantine, Computational and Algebraic Aspects: Proceedings of the International Conference held in Eger, Hungary. Walter de Grutyer and Co. (1998), 371-382.
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| ==Generalizations==
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| [[Automorphic form]]s generalize the idea of modular forms for general [[Lie group]]s; and Eisenstein series generalize in a similar fashion.
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| Defining ''O<sub>K</sub>'' to be the [[ring of integers]] of a [[totally real algebraic number field]] K, one then defines the [[Hilbert-Blumenthal modular group]] as PSL(2,''O<sub>K</sub>''). One can then associate an Eisenstein series to every [[cusp form|cusp]] of the Hilbert-Blumenthal modular group.
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| == References ==
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| * Naum Illyich Akhiezer, ''Elements of the Theory of Elliptic Functions'', (1970) Moscow, translated into English as ''AMS Translations of Mathematical Monographs Volume 79'' (1990) AMS, Rhode Island ISBN 0-8218-4532-2
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| * Tom M. Apostol, ''Modular Functions and Dirichlet Series in Number Theory, Second Edition'' (1990), Springer, New York ISBN 0-387-97127-0
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| * [[Henryk Iwaniec]], ''Spectral Methods of Automorphic Forms, Second Edition'', (2002) (Volume 53 in ''Graduate Studies in Mathematics''), America Mathematical Society, Providence, RI ISBN 0-8218-3160-7 ''(See chapter 3)''
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| * [[Jean-Pierre Serre|Serre, Jean-Pierre]], ''A course in arithmetic''. Translated from the French. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973.
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| [[Category:Mathematical series]]
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| [[Category:Modular forms]]
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| [[Category:Analytic number theory]]
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| [[Category:Fractals]]
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