Errata to the papers : “Connections between <span class="mathjax-formula">$B_{2,\chi }$</span> for even quadratic Dirichlet characters <span class="mathjax-formula">$\chi $</span>” and “Class numbers of appropriate imaginary quadratic fields, parts I and

C ^OMPOSITIO M ^ATHEMATICA

J ^ERZY U ^RBANOWICZ

Errata to the papers : “Connections between B

for even quadratic Dirichlet characters χ”

and “Class numbers of appropriate imaginary quadratic fields, parts I and II”

Compositio Mathematica, tome 77, n^o1 (1991), p. 119-125

<http://www.numdam.org/item?id=CM_1991__77_1_119_0>

© Foundation Compositio Mathematica, 1991, tous droits réservés.

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x

appropriate imaginary quadratic

(published

Compositio

1990)

JERZY URBANOWICZ

Institute of Mathematics, ^Polish Academy of Sciences, ^ul. Sniadeckich 8, 00-950 Warszawa, Poland

Due to a printer’s error, 74 pairs ^of parentheses in Kronecker symbols ^were omitted, ^which completely changed ^{the sense of the} theorems of both _papers.

Here are their correct versions.

Part I:

THEOREM 1. Let for k ⁼ 0,1, 2 ^{and 3}

120

COROLLARY 1. Let ₉ denote Euler’s totient function.

(i) k2(D) ⁼ 2h( - 4D) ⁺ 2cp(D) ⁺ 03B5 (mod 32),

where e ⁼ 0 unless D ⁼ p ^{~ -} 3(mod 8) ^a prime ^{or D =} pq, where _p ^~ q ~ 1(mod 8)

or p==q+4==3(mod8), p, q-primes. ^{In these cases 03B5 = 16} if p ~ q ~ 3(mod 8),

8 = - 8 if p ^~ q ~ - 1(mod 8) ^{and e = 8 otherwise.}

p - 1(mod 8), p - 5(mod 16), resp. p ^{~ -} 3(mod 16),

where a ⁼ 0 if p ^--- 1(mod 16) ^{and a = 1} otherwise, ^and fi = - 1, - 3, resp. 5 if

p == 1(mod 8), p - - 3(mod 16), resp. p - 5(mod 16).

THEOREM 2. Let for k ⁼ 0,1, 2 ^{and 3}

Then for ^{0394 ~ 3:}

122

COROLLARY 1.

where e=0 unless A ⁼ _p ^--- 3(mod 4) a prime ^{or A = pq, where p = q + 2 -} -1(mod 8), p, q-primes, ^{or A = pqr, where p = q ~ r ~ -1,} 3(mod 8), ^or p - q

== -1, resp. 3(mod 8) ^{and r -} 3, resp. -1(mod 8), ^{p, q,} r-primes.

1 n these cases 03B5 = 4 if 0394 = p ~ 1(mod 8), 03B5 = - 4 if 0394 = p ~ 3(mod 8) ^and

G ⁼ 16 otherwise.

k2(80394) ⁼ 2h( - 8A))(mod 32), if ^A ~ -1(mod 8), ⁱⁿ particular,

where a ⁼ 1 if p ^~ 7(mod 16) ^{and a = 0} otherwise, ^and fi = -1, 3, resp. 11 if

p ~ -1(mod 8), p - 3(mod 16), resp. p - 11 (mod 16).

Part II:

LEMMA 1 ([5], [1]). ^{We have:}

where À(e) ^{= 1,} if ^{e = -} 3, ^and À(e) ⁼ 0, ^otherwise.

Moreover ^we have for ^{k =} 5, 6, 7, ⁸

124

LEMMA 2 ([8]). ^{We have:}