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Sieve methods and class-number problems. II. (English) Zbl 0665.10029

The authors continue their study on the connections between sieve upper bounds for polynomials \(f_ k=f_ k(n,d)\) in two variables n, d and weighted sums of \(L(1,\chi)^{-k},\quad k\in {\mathbb{N}}.\)
In contrast to Part I [ibid. 367, 11-26 (1986; Zbl 0588.10053)], this Part II is mainly concerned with asymptotic formulas for \(\sum_{d\leq y}c_ k(d)L(1,\chi)^{-k}\) with ineffective (Theorem 1) and effective (Theorem 2, \(k=2)\) error estimates, and the sieve upper bound for \(S_ f(x,y,z)\) (counting the number of pairs (n,d) with \(n\leq x\), \(d\leq y\) such that \(f_ k\) is coprime to all the primes up to z) (Theorem 3) is an aid for obtaining a hypothetical effective version of Siegel’s theorem (the hypothesis H(A) refers to the uniform effective error estimate in the prime number theorem for arithmetic progression mod D with \(D\ll (\log t)^{A-\epsilon}.\)
Instead of the single polynomial \(n^ 2+d\), the authors consider their product \[ f_ k(n,d)=\prod^{k-1}_{i=0}(n^ 2+a_ i^ 2d),\quad a_ 0=1. \] With \(f_ k\) associated are the functions \(\rho_{d,f_ k}(p)\) (resp. \(\omega_{n,f_ k}(p))\) which count the number of solutions n mod p (resp. d mod p) of \(f_ k\equiv 0(mod p)\). The coefficients \(a_ i\) (being equal to the product of first \(2i+2\) primes- 1) are chosen so that both \(\rho\), \(\omega\) are \(<p.\)
The proofs of Theorems 1 and 2 go along the similar lines to those of the proof of Theorem 2 in Part I, with considerable complications to deal with possible quadratic factors of \(f_ k\). If the p’s are restricted, \(\rho_{d,f_ k}\) reduces to \(\rho_ d\) in Part I and \(\omega_{n,f_ k}\), to k, and similar reduction can be made as in Part I. Ineffectiveness and effectiveness arises according as Siegel’s theorem or the Goldfeld-Gross-Zagier theorem (plus Tatuzawa’s theorem) is used.
Reviewer: S.Kanemitsu

MSC:

11N35 Sieves
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11R23 Iwasawa theory
11R11 Quadratic extensions

Citations:

Zbl 0588.10053
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