Efficient computation of the Euler–Kronecker constants of prime cyclotomic fields

Alessandro Languasco

摘要

We introduce a new algorithm, which is faster and requires less computing resources than the ones previously known, to compute the Euler–Kronecker constants G q documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathfrak {G}}_q$$end{document} for the prime cyclotomic fields Q ( ζ q ) documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ {mathbb {Q}}(zeta _q)$$end{document} , where q is an odd prime and ζ q documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$zeta _q$$end{document} is a primitive q -root of unity. With such a new algorithm we evaluated G q documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathfrak {G}}_q$$end{document} and G q + documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathfrak {G}}_q^+$$end{document} , where G q + documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathfrak {G}}_q^+$$end{document} is the Euler–Kronecker constant of the maximal real subfield of Q ( ζ q ) documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathbb {Q}}(zeta _q)$$end{document} , for some very large primes q thus obtaining two new negative values of G q documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathfrak {G}}_q$$end{document} : G 9109334831 = - 0.248739 ? documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathfrak {G}}_{9109334831}= -0.248739dotsc $$end{document} and G 9854964401 = - 0.096465 ? documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathfrak {G}}_{9854964401}= -0.096465dotsc $$end{document} We also evaluated G q documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathfrak {G}}_q$$end{document} and G q + documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathfrak {G}}^+_q$$end{document} for every odd prime q ≤ 10 6 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$qle 10^6$$end{document} , thus enlarging the size of the previously known range for G q documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathfrak {G}}_q$$end{document} and G q + documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathfrak {G}}^+_q$$end{document} . Our method also reveals that the difference G q - G q + documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${ma

机译：我们介绍了一种新的算法，该算法比先前已知的速度更快，需要较少的计算资源，以计算euler-kronecker常量g q documentclass [12pt] {minimal} usepackage {ammath} usepackage {keysym} usepackage { Amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs} usepackage {supmeek} setLength { oddsideDemargin} {-69pt} begin {document} $$ { mathfrak {g}} _q $$结束{document}对于Prime Carchotomic字段q（ζq） documentclass [12pt] {minimal} usepackage {ammath} usepackage {isysym} usepackage {amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage { mathrsfs} usepackage {supmeek} setLength { oddsidemargin} { - 69pt} begin {document} $$ { mathbb {q}}（ zeta _q）$$ end {document}，其中q是一个奇数和ζq documentclass [12pt] {minimal} usepackage {ammath} usepackage {isysym} usepackage {amsfonts} usepackage {amssys} usepackage {amsbsy} usepackage {mathrsfs} usepackage {supmeek} setLength { Oddsidemargin} { - 69pt} begin {document} $$ zeta _q $$$ end {document}是一个原始的q -root的团结。使用这种新算法，我们评估了g q documentclass [12pt] {minimal} usepackage {ammath} usepackage {isysym} usepackage {amsfonts} usepackage {amsbsy} usepackage {mathrsfs} usepackage {升级} setLength { oddsidemargin} { - 69pt} begin {document} $$ { mathfrak {g}} _ q $$ neg {document}和g q + documentclass [12pt] {minimal} usepackage {ammath } usepackage {isysym} usepackage {amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs} usepackage {supmeek} setLength { oddsidemargin} { - 69pt} begin {document} $$ { mathfrak {g}} _ q ^ + $$$ nocopy {document}，其中g + documentclass [12pt] {minimal} usepackage {ammath} usepackage {keysym} usepackage {amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs} usepackage {supmeek} setLength { oddsidemargin} { - 69pt} begin {document} $$ { mathfrak {g}} _ q ^ + $$} _ q ^ + $$$} _ q ^ + $$} _ q ^ + $$} _ q ^ + $$} _ q ^ + $$$$ {document}是euler -kroncrecker q（ζq） documentclass [12pt] {minimal} usepackage {ammath} usepackage { usysym} usepackage {amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs} usepackage {matheek} setLength { oddsideDemargin} { - 69pt} begin {document} $$ { mathbb {q} }（ zeta _q）$$ end {document}，对于一些非常大的primes q因此获得了g q documentClass [12pt]的两个新负值[12pt] {minimal} usepackage {ammath} usepackage {isysym} usepackage { Amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs} usepackage {supmeek} setLength { oddsideDemargin} {-69pt} begin {document} $$ { mathfrak {g}} _q $$结束{document}：g 9109334831 = - 0.248739？ DocumentClass [12pt] {minimal} usepackage {ammath} usepackage {keysym} usepackage {amsfonts} usepackage {amssysfs} usepackage {mathrsfs} usepackage {supmeek} setLength { oddsidemargin} { -69pt} begin {document} $$ { mathfrak {g}} _ {9109334831} = {9109334831} = {9109334831} = {9109334831} = {9109334831} = {9109334831} = -0.248739 dotsc $$ end {document}和g 9854964401 = - 0.096465？ DocumentClass [12pt] {minimal} usepackage {ammath} usepackage {keysym} usepackage {amsfonts} usepackage {amssysfs} usepackage {mathrsfs} usepackage {supmeek} setLength { oddsidemargin} { -69pt} {开始文档} $$ { mathfrak {G}} _ {9854964401} = -0.096465 dotsc $$ {端文档}我们也评估ģq 的DocumentClass [12磅] {最小} {usepackage amsmath } usepackage {isysym} usepackage {amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs} usepackage {supmeek} setLength { oddsidemargin} { - 69pt} begin {document} $$ { mathfrak {g}} _ q $$ end {document}和g q + documentclass [12pt] {minimal} usepackage {ammath} usepackage {isysym} usepackage {amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs} {usepackage upgreek} setlength { oddsidemargin} { - 69pt} {开始文档} $$ { mathfrak {G}} ^ + _ q $$ {端文档}对于每个奇素数q≤ 10 6 documentClass [12pt] {minimal} usepackage {ammath} usepackage {isysym} usepackage {amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs} usepackage {supmeek} setLength { oddsidemargin} { - 69pt} begin {document} $$ q le 10 ^ 6 $$ end {document}，从而扩大大小先前已知的g documentClass [12pt] {minimal} usepackage {ammath} usepackage {isysym} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs} usepackage {mathrsfs} usepackage {mathrsfs} setLength { oddsidemargin} { - 69pt} begin {document} $$ { mathfrak {g}} _ q $$ neg {document}和g q + documentclass [12pt] {minimal} usepackage {ammath} usepackage {isysym} usepackage {amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs} usepackage {supmeek} setLength { oddsideDemargin} {-69pt} begin {document} $$ { mathfrak { g}} ^ + _ q $$ 结束{document}。我们的方法还揭示了差异g q-g q + documentClass [12pt] {minimal} usepackage {ammath} usepackage {keysym} usepackage {amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs } usepackage {supmeez} setLength { oddsidemargin} { - 69pt} begin {document} $$ { ma

Efficient computation of the Euler–Kronecker constants of prime cyclotomic fields

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