Euler product




In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function.




Contents






  • 1 Definition


  • 2 Convergence


  • 3 Examples


  • 4 Notable constants


  • 5 Notes


  • 6 References


  • 7 External links





Definition


In general, if a{displaystyle a}a is a multiplicative function, then the Dirichlet series


∑na(n)n−s{displaystyle sum _{n}a(n)n^{-s},}sum _{n}a(n)n^{-s},

is equal to


∏pP(p,s){displaystyle prod _{p}P(p,s),}prod _{p}P(p,s),

where the product is taken over prime numbers p{displaystyle p}p, and P(p,s){displaystyle P(p,s)}P(p,s) is the sum


1+a(p)p−s+a(p2)p−2s+⋯.{displaystyle 1+a(p)p^{-s}+a(p^{2})p^{-2s}+cdots .}1+a(p)p^{-s}+a(p^{2})p^{-2s}+cdots .

In fact, if we consider these as formal generating functions, the existence of such a formal Euler product expansion is a necessary and sufficient condition that a(n){displaystyle a(n)}a(n) be multiplicative: this says exactly that a(n){displaystyle a(n)}a(n) is the product of the a(pk){displaystyle a(p^{k})}a(p^{k}) whenever n{displaystyle n}n factors as the product of the powers pk{displaystyle p^{k}}p^{k} of distinct primes p{displaystyle p}p.


An important special case is that in which a(n){displaystyle a(n)}a(n) is totally multiplicative, so that P(p,s){displaystyle P(p,s)}P(p,s) is a geometric series. Then


P(p,s)=11−a(p)p−s,{displaystyle P(p,s)={frac {1}{1-a(p)p^{-s}}},}P(p,s)={frac {1}{1-a(p)p^{-s}}},

as is the case for the Riemann zeta-function, where a(n)=1{displaystyle a(n)=1}a(n)=1, and more generally for Dirichlet characters.



Convergence


In practice all the important cases are such that the infinite series and infinite product expansions are absolutely convergent in some region


Re⁡(s)>C,{displaystyle operatorname {Re} (s)>C,}{displaystyle operatorname {Re} (s)>C,}

that is, in some right half-plane in the complex numbers. This already gives some information, since the infinite product, to converge, must give a non-zero value; hence the function given by the infinite series is not zero in such a half-plane.


In the theory of modular forms it is typical to have Euler products with quadratic polynomials in the denominator here. The general Langlands philosophy includes a comparable explanation of the connection of polynomials of degree m, and the representation theory for GLm.



Examples


The Euler product attached to the Riemann zeta function ζ(s),{displaystyle zeta (s),}{displaystyle zeta (s),} using also the sum of the geometric series, is


∏p(1−p−s)−1=∏p(∑n=0∞p−ns)=∑n=1∞1ns=ζ(s).{displaystyle prod _{p}(1-p^{-s})^{-1}=prod _{p}{Big (}sum _{n=0}^{infty }p^{-ns}{Big )}=sum _{n=1}^{infty }{frac {1}{n^{s}}}=zeta (s).}{displaystyle prod _{p}(1-p^{-s})^{-1}=prod _{p}{Big (}sum _{n=0}^{infty }p^{-ns}{Big )}=sum _{n=1}^{infty }{frac {1}{n^{s}}}=zeta (s).}

while for the Liouville function λ(n)=(−1)Ω(n),{displaystyle lambda (n)=(-1)^{Omega (n)},}{displaystyle lambda (n)=(-1)^{Omega (n)},} it is


∏p(1+p−s)−1=∑n=1∞λ(n)ns=ζ(2s)ζ(s).{displaystyle prod _{p}(1+p^{-s})^{-1}=sum _{n=1}^{infty }{frac {lambda (n)}{n^{s}}}={frac {zeta (2s)}{zeta (s)}}.}{displaystyle prod _{p}(1+p^{-s})^{-1}=sum _{n=1}^{infty }{frac {lambda (n)}{n^{s}}}={frac {zeta (2s)}{zeta (s)}}.}

Using their reciprocals, two Euler products for the Möbius function μ(n){displaystyle mu (n)}mu (n) are


∏p(1−p−s)=∑n=1∞μ(n)ns=1ζ(s){displaystyle prod _{p}(1-p^{-s})=sum _{n=1}^{infty }{frac {mu (n)}{n^{s}}}={frac {1}{zeta (s)}}}prod _{p}(1-p^{-s})=sum _{n=1}^{infty }{frac {mu (n)}{n^{s}}}={frac {1}{zeta (s)}}

and


∏p(1+p−s)=∑n=1∞|μ(n)|ns=ζ(s)ζ(2s).{displaystyle prod _{p}(1+p^{-s})=sum _{n=1}^{infty }{frac {|mu (n)|}{n^{s}}}={frac {zeta (s)}{zeta (2s)}}.}{displaystyle prod _{p}(1+p^{-s})=sum _{n=1}^{infty }{frac {|mu (n)|}{n^{s}}}={frac {zeta (s)}{zeta (2s)}}.}

Taking the ratio of these two gives


∏p(1+p−s1−p−s)=∏p(ps+1ps−1)=ζ(s)2ζ(2s).{displaystyle prod _{p}{Big (}{frac {1+p^{-s}}{1-p^{-s}}}{Big )}=prod _{p}{Big (}{frac {p^{s}+1}{p^{s}-1}}{Big )}={frac {zeta (s)^{2}}{zeta (2s)}}.}{displaystyle prod _{p}{Big (}{frac {1+p^{-s}}{1-p^{-s}}}{Big )}=prod _{p}{Big (}{frac {p^{s}+1}{p^{s}-1}}{Big )}={frac {zeta (s)^{2}}{zeta (2s)}}.}

Since for even s the Riemann zeta function ζ(s){displaystyle zeta (s)}zeta (s) has an analytic expression in terms of a rational multiple of πs,{displaystyle pi ^{s},}{displaystyle pi ^{s},} then for even exponents, this infinite product evaluates to a rational number. For example, since ζ(2)=π2/6,{displaystyle zeta (2)=pi ^{2}/6,}{displaystyle zeta (2)=pi ^{2}/6,} ζ(4)=π4/90,{displaystyle zeta (4)=pi ^{4}/90,}zeta (4)=pi ^{4}/90, and ζ(8)=π8/9450,{displaystyle zeta (8)=pi ^{8}/9450,}{displaystyle zeta (8)=pi ^{8}/9450,} then


∏p(p2+1p2−1)=52,{displaystyle prod _{p}{Big (}{frac {p^{2}+1}{p^{2}-1}}{Big )}={frac {5}{2}},}{displaystyle prod _{p}{Big (}{frac {p^{2}+1}{p^{2}-1}}{Big )}={frac {5}{2}},}

∏p(p4+1p4−1)=76,{displaystyle prod _{p}{Big (}{frac {p^{4}+1}{p^{4}-1}}{Big )}={frac {7}{6}},}{displaystyle prod _{p}{Big (}{frac {p^{4}+1}{p^{4}-1}}{Big )}={frac {7}{6}},}

and so on, with the first result known by Ramanujan. This family of infinite products is also equivalent to


∏p(1+2p−s+2p−2s+⋯)=∑n=1∞2ω(n)n−s=ζ(s)2ζ(2s),{displaystyle prod _{p}(1+2p^{-s}+2p^{-2s}+cdots )=sum _{n=1}^{infty }2^{omega (n)}n^{-s}={frac {zeta (s)^{2}}{zeta (2s)}},}{displaystyle prod _{p}(1+2p^{-s}+2p^{-2s}+cdots )=sum _{n=1}^{infty }2^{omega (n)}n^{-s}={frac {zeta (s)^{2}}{zeta (2s)}},}

where ω(n){displaystyle omega (n)}omega (n) counts the number of distinct prime factors of n, and 2ω(n){displaystyle 2^{omega (n)}}2^{omega (n)} is the number of square-free divisors.


If χ(n){displaystyle chi (n)}chi (n) is a Dirichlet character of conductor N,{displaystyle N,}N, so that χ{displaystyle chi }chi is totally multiplicative and χ(n){displaystyle chi (n)}chi (n) only depends on n modulo N, and χ(n)=0{displaystyle chi (n)=0}chi (n)=0 if n is not coprime to N, then


∏p(1−χ(p)p−s)−1=∑n=1∞χ(n)n−s.{displaystyle prod _{p}(1-chi (p)p^{-s})^{-1}=sum _{n=1}^{infty }chi (n)n^{-s}.}{displaystyle prod _{p}(1-chi (p)p^{-s})^{-1}=sum _{n=1}^{infty }chi (n)n^{-s}.}

Here it is convenient to omit the primes p dividing the conductor N from the product. In his notebooks, Ramanujan generalized the Euler product for the zeta function as


∏p(x−p−s)≈1Lis⁡(x){displaystyle prod _{p}(x-p^{-s})approx {frac {1}{operatorname {Li} _{s}(x)}}}prod _{p}(x-p^{-s})approx {frac {1}{operatorname {Li} _{s}(x)}}

for s>1{displaystyle s>1}s>1 where Lis⁡(x){displaystyle operatorname {Li} _{s}(x)}operatorname {Li} _{s}(x) is the polylogarithm. For x=1{displaystyle x=1}x=1 the product above is just 1/ζ(s).{displaystyle 1/zeta (s).}1/zeta (s).



Notable constants


Many well known constants have Euler product expansions.


The Leibniz formula for π,


π4=∑n=0∞(−1)n2n+1=1−13+15−17+⋯,{displaystyle {frac {pi }{4}}=sum _{n=0}^{infty }{frac {(-1)^{n}}{2n+1}}=1-{frac {1}{3}}+{frac {1}{5}}-{frac {1}{7}}+cdots ,}{displaystyle {frac {pi }{4}}=sum _{n=0}^{infty }{frac {(-1)^{n}}{2n+1}}=1-{frac {1}{3}}+{frac {1}{5}}-{frac {1}{7}}+cdots ,}

can be interpreted as a Dirichlet series using the (unique) Dirichlet character modulo 4, and converted to an Euler product of superparticular ratios


π4=(∏p≡1(mod4)pp−1)⋅(∏p≡3(mod4)pp+1)=34⋅54⋅78⋅1112⋅1312⋯,{displaystyle {frac {pi }{4}}=left(prod _{pequiv 1{pmod {4}}}{frac {p}{p-1}}right)cdot left(prod _{pequiv 3{pmod {4}}}{frac {p}{p+1}}right)={frac {3}{4}}cdot {frac {5}{4}}cdot {frac {7}{8}}cdot {frac {11}{12}}cdot {frac {13}{12}}cdots ,}{displaystyle {frac {pi }{4}}=left(prod _{pequiv 1{pmod {4}}}{frac {p}{p-1}}right)cdot left(prod _{pequiv 3{pmod {4}}}{frac {p}{p+1}}right)={frac {3}{4}}cdot {frac {5}{4}}cdot {frac {7}{8}}cdot {frac {11}{12}}cdot {frac {13}{12}}cdots ,}

where each numerator is a prime number and each denominator is the nearest multiple of four.[1]


Other Euler products for known constants include:


Hardy–Littlewood's twin prime constant:


∏p>2(1−1(p−1)2)=0.660161...{displaystyle prod _{p>2}left(1-{frac {1}{(p-1)^{2}}}right)=0.660161...}{displaystyle prod _{p>2}left(1-{frac {1}{(p-1)^{2}}}right)=0.660161...}

Landau-Ramanujan constant:


π4∏p≡1(mod4)(1−1p2)1/2=0.764223...{displaystyle {frac {pi }{4}}prod _{pequiv 1{pmod {4}}}left(1-{frac {1}{p^{2}}}right)^{1/2}=0.764223...}{displaystyle {frac {pi }{4}}prod _{pequiv 1{pmod {4}}}left(1-{frac {1}{p^{2}}}right)^{1/2}=0.764223...}

12∏p≡3(mod4)(1−1p2)−1/2=0.764223...{displaystyle {frac {1}{sqrt {2}}}prod _{pequiv 3{pmod {4}}}left(1-{frac {1}{p^{2}}}right)^{-1/2}=0.764223...}{displaystyle {frac {1}{sqrt {2}}}prod _{pequiv 3{pmod {4}}}left(1-{frac {1}{p^{2}}}right)^{-1/2}=0.764223...}

Murata's constant (sequence A065485 in the OEIS):


∏p(1+1(p−1)2)=2.826419...{displaystyle prod _{p}left(1+{frac {1}{(p-1)^{2}}}right)=2.826419...}{displaystyle prod _{p}left(1+{frac {1}{(p-1)^{2}}}right)=2.826419...}

Strongly carefree constant ×ζ(2)2{displaystyle times zeta (2)^{2}}times zeta (2)^{2} OEIS: A065472:


∏p(1−1(p+1)2)=0.775883...{displaystyle prod _{p}left(1-{frac {1}{(p+1)^{2}}}right)=0.775883...}{displaystyle prod _{p}left(1-{frac {1}{(p+1)^{2}}}right)=0.775883...}

Artin's constant OEIS: A005596:


∏p(1−1p(p−1))=0.373955...{displaystyle prod _{p}left(1-{frac {1}{p(p-1)}}right)=0.373955...}{displaystyle prod _{p}left(1-{frac {1}{p(p-1)}}right)=0.373955...}

Landau's totient constant OEIS: A082695:


∏p(1+1p(p−1))=3152π4ζ(3)=1.943596...{displaystyle prod _{p}left(1+{frac {1}{p(p-1)}}right)={frac {315}{2pi ^{4}}}zeta (3)=1.943596...}{displaystyle prod _{p}left(1+{frac {1}{p(p-1)}}right)={frac {315}{2pi ^{4}}}zeta (3)=1.943596...}

Carefree constant ×ζ(2){displaystyle times zeta (2)}times zeta (2) OEIS: A065463:


∏p(1−1p(p+1))=0.704442...{displaystyle prod _{p}left(1-{frac {1}{p(p+1)}}right)=0.704442...}{displaystyle prod _{p}left(1-{frac {1}{p(p+1)}}right)=0.704442...}

(with reciprocal) OEIS: A065489:


∏p(1+1p2+p−1)=1.419562...{displaystyle prod _{p}left(1+{frac {1}{p^{2}+p-1}}right)=1.419562...}{displaystyle prod _{p}left(1+{frac {1}{p^{2}+p-1}}right)=1.419562...}

Feller-Tornier constant OEIS: A065493:


12+12∏p(1−2p2)=0.661317...{displaystyle {frac {1}{2}}+{frac {1}{2}}prod _{p}left(1-{frac {2}{p^{2}}}right)=0.661317...}{displaystyle {frac {1}{2}}+{frac {1}{2}}prod _{p}left(1-{frac {2}{p^{2}}}right)=0.661317...}

Quadratic class number constant OEIS: A065465:


∏p(1−1p2(p+1))=0.881513...{displaystyle prod _{p}left(1-{frac {1}{p^{2}(p+1)}}right)=0.881513...}{displaystyle prod _{p}left(1-{frac {1}{p^{2}(p+1)}}right)=0.881513...}

Totient summatory constant OEIS: A065483:


∏p(1+1p2(p−1))=1.339784...{displaystyle prod _{p}left(1+{frac {1}{p^{2}(p-1)}}right)=1.339784...}{displaystyle prod _{p}left(1+{frac {1}{p^{2}(p-1)}}right)=1.339784...}

Sarnak's constant OEIS: A065476:


∏p>2(1−p+2p3)=0.723648...{displaystyle prod _{p>2}left(1-{frac {p+2}{p^{3}}}right)=0.723648...}{displaystyle prod _{p>2}left(1-{frac {p+2}{p^{3}}}right)=0.723648...}

Carefree constant OEIS: A065464:


∏p(1−2p−1p3)=0.428249...{displaystyle prod _{p}left(1-{frac {2p-1}{p^{3}}}right)=0.428249...}{displaystyle prod _{p}left(1-{frac {2p-1}{p^{3}}}right)=0.428249...}

Strongly carefree constant OEIS: A065473:


∏p(1−3p−2p3)=0.286747...{displaystyle prod _{p}left(1-{frac {3p-2}{p^{3}}}right)=0.286747...}{displaystyle prod _{p}left(1-{frac {3p-2}{p^{3}}}right)=0.286747...}

Stephens' constant OEIS: A065478:


∏p(1−pp3−1)=0.575959...{displaystyle prod _{p}left(1-{frac {p}{p^{3}-1}}right)=0.575959...}{displaystyle prod _{p}left(1-{frac {p}{p^{3}-1}}right)=0.575959...}

Barban's constant OEIS: A175640:


∏p(1+3p2−1p(p+1)(p2−1))=2.596536...{displaystyle prod _{p}left(1+{frac {3p^{2}-1}{p(p+1)(p^{2}-1)}}right)=2.596536...}{displaystyle prod _{p}left(1+{frac {3p^{2}-1}{p(p+1)(p^{2}-1)}}right)=2.596536...}

Taniguchi's constant OEIS: A175639:


∏p(1−3p3+2p4+1p5−1p6)=0.678234...{displaystyle prod _{p}left(1-{frac {3}{p^{3}}}+{frac {2}{p^{4}}}+{frac {1}{p^{5}}}-{frac {1}{p^{6}}}right)=0.678234...}{displaystyle prod _{p}left(1-{frac {3}{p^{3}}}+{frac {2}{p^{4}}}+{frac {1}{p^{5}}}-{frac {1}{p^{6}}}right)=0.678234...}

Heath-Brown and Moroz constant OEIS: A118228:


∏p(1−1p)7(1+7p+1p2)=0.0013176...{displaystyle prod _{p}left(1-{frac {1}{p}}right)^{7}left(1+{frac {7p+1}{p^{2}}}right)=0.0013176...}{displaystyle prod _{p}left(1-{frac {1}{p}}right)^{7}left(1+{frac {7p+1}{p^{2}}}right)=0.0013176...}


Notes





  1. ^ Debnath, Lokenath (2010), The Legacy of Leonhard Euler: A Tricentennial Tribute, World Scientific, p. 214, ISBN 9781848165267.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.




References




  • G. Polya, Induction and Analogy in Mathematics Volume 1 Princeton University Press (1954) L.C. Card 53-6388 (A very accessible English translation of Euler's memoir regarding this "Most Extraordinary Law of the Numbers" appears starting on page 91)


  • Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
    (Provides an introductory discussion of the Euler product in the context of classical number theory.)


  • G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, 5th ed., Oxford (1979)
    ISBN 0-19-853171-0 (Chapter 17 gives further examples.)

  • George E. Andrews, Bruce C. Berndt, Ramanujan's Lost Notebook: Part I, Springer (2005),
    ISBN 0-387-25529-X

  • G. Niklasch, Some number theoretical constants: 1000-digit values"



External links




  • "Euler product". PlanetMath.


  • Hazewinkel, Michiel, ed. (2001) [1994], "Euler product", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

  • Weisstein, Eric W. "Euler Product". MathWorld.


  • Niklasch, G. (23 Aug 2002). "Some number-theoretical constants". Archived from the original on 12 Jun 2006.




Comments

Post a Comment

Popular posts from this blog

Information security

Image
This article is part of a series on Information security Related security categories Internet security Cyberwarfare Computer security Mobile security Network security Threats Computer crime Vulnerability Eavesdropping Malware Spyware Ransomware Trojans Viruses Worms Rootkits Bootkits Keyloggers Screen scrapers Exploits Backdoors Logic bombs Payloads Denial of service Defenses Computer access control Application security Antivirus software Secure coding Secure by default Secure by design Secure operating systems Authentication Multi-factor authentication Authorization Data-centric security Encryption Firewall Intrusion detection system Mobile secure gateway Runtime application self-protection (RASP) v t e Information security , sometimes shortened to InfoSec , is the practice of preventing unauthorized access, use, disclosure, disruption, modification, inspection, recording or destruction of information. Th...
Read more

Monte Carlo

Image
For other uses, see Monte Carlo (disambiguation). Quarter and ward of Monaco in Monaco Monte Carlo Quarter and ward of Monaco Monte Carlo in Monaco (Ward shown) Monte Carlo Location in relation to France Coordinates: 43°44′23″N 7°25′38″E  /  43.73972°N 7.42722°E  / 43.73972; 7.42722 Coordinates: 43°44′23″N 7°25′38″E  /  43.73972°N 7.42722°E  / 43.73972; 7.42722 Country   Monaco Government  • Type Principality Area  • Urban 0.61 km 2 (.234 sq mi) Population  • Quarter and ward of Monaco 15,200 (in the quarter)  3,500 (in the ward) Postcode 98000 Monte Carlo ( / ˈ m ɒ n t i ˈ k ɑːr l oʊ / ; Italian:  [ˈmonte ˈkarlo] ; French: Monte-Carlo [mɔ̃te kaʁlo] , or colloquially Monte-Carl [mɔ̃te kaʁl] ; Monégasque: Munte Carlu ) officially refers to an administrative area of the Principality of Monaco, specifically the ward of Monte Carlo/Spélugues, where the Monte Carlo Casino is located. Informally the name also refers to a ...
Read more

孟学农

Image
body.skin-minerva .mw-parser-output table.infobox caption{text-align:center} 孟学农 北京市人民政府市长 任期 2003年1月 － 4月 山西省人民政府省长 任期 2007年9月 － 2008年9月 个人资料 性别 男 出生 1949年8月（69歲）   中華民國山东省蓬莱县 籍贯 中國山东 政党  中国共产党 学历 中国人民大学马列主义专业 工学硕士 经历 北京市副市长 (1993年2月–2003年1月) 全国人大澳门特别行政区筹委会委员 (1998年5月–1999年2月) 中共中央委员 (2002年11月–至今) 中共北京市委副书记 (2002年12月–2003年4月) 北京市市长 (2003年1月–4月) 国务院南水北调工程建设委员会办公室副主任 (2003年9月–2007年7月) 中共山西省委副书记 (2007年8月–2008年9月) 山西省省长 (2007年9月–2008年9月) 中央直属机关工委副书记 (2010年1月至今) 第十二届全国政协常务委员 (2013年3月－2018年3月) 孟学农 （1949年8月 － ），中国山东省蓬莱县人；毕业于中国人民大学马列主义专业，工学硕士。1972年7月加入中国共产党；是中共第十六、十七、十八届中央委员。现任第十二届全国政协常务委员会委员、全国政协社会和法制委员会主任。 曾在2003年1月当选北京市市长；三个月后，因处理SARS事件不力，引咎辞职。2007年9月，出任山西省省長；一年后，又因襄汾尾矿库溃坝事故，再度引咎辞职。 目录 1 从政经历 1.1 “短命”市长 1.2 东山再起 1.3 二度复出 2 家族关系 3 参考文献 从政经历 1968年高中毕业后的孟学农，没有像当时多数同龄人一样“上山下乡”，而是进入北京第二汽车制造厂当工人。在工作期间，孟学农加入了中国共产党，还担任了该厂共青团委书记。而后，他又被北京团市委看中，于1970年代末期曾调入共青团北京市委工交部工作，开始...
Read more