An explicit upper bound for $$L(1,\chi )$$ when $$\chi $$ is quadratic

Pub Date : 2023-10-03 DOI:10.1007/s40993-023-00476-4

D. R. Johnston, O. Ramaré, T. Trudgian

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引用次数: 0

Abstract

Abstract We consider Dirichlet L -functions $$L(s, \chi )$$ L ( s , χ ) where $$\chi $$ χ is a non-principal quadratic character to the modulus q . We make explicit a result due to Pintz and Stephens by showing that $$|L(1, \chi )|\leqslant \frac{1}{2}\log q$$ | L ( 1 , χ ) | ⩽ 1 2 log q for all $$q\geqslant 2\cdot 10^{23}$$ q ⩾ 2 · 10 23 and $$|L(1, \chi )|\leqslant \frac{9}{20}\log q$$ | L ( 1 , χ ) | ⩽ 9 20 log q for all $$q\geqslant 5\cdot 10^{50}$$ q ⩾ 5 · 10 50 .

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当$$\chi $$是二次元时，$$L(1,\chi )$$的显式上界

我们考虑Dirichlet L -函数$$L(s, \chi )$$ L (s， χ)，其中$$\chi $$ χ是模q的非主二次特征。我们通过显示所有$$q\geqslant 2\cdot 10^{23}$$ q大于或等于2·10 23的$$|L(1, \chi )|\leqslant \frac{1}{2}\log q$$ | L (1， χ) |≤1 2 log q和所有$$q\geqslant 5\cdot 10^{50}$$ q大于或等于5·10 50的$$|L(1, \chi )|\leqslant \frac{9}{20}\log q$$ | L (1， χ) |≤9 20 log q来明确pinz和Stephens的结果。

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