Journal of the Ramanujan Mathematical Society

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A Short Note on the Divisibility of Class Numbers of Real Quadratic Fields


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1 Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India
     

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For any integer l ≥ 1, let p1, p2, . . . , pl+2 be distinct prime numbers ≥ 5. For all real numbers X > 1, we let N3,l (X) denote the number of real quadratic fields K whose absolute discriminant dK ≤ X and dK is divisible by (p1 . . . pl+2) together with the class number hK of K divisible by 2l · 3. Then, in this short note, by following the method in [3], we prove that N3,l (X) ≫ X 7/8 for all large enough X’s.
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  • A Short Note on the Divisibility of Class Numbers of Real Quadratic Fields

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Authors

Jaitra Chattopadhyay
Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India

Abstract


For any integer l ≥ 1, let p1, p2, . . . , pl+2 be distinct prime numbers ≥ 5. For all real numbers X > 1, we let N3,l (X) denote the number of real quadratic fields K whose absolute discriminant dK ≤ X and dK is divisible by (p1 . . . pl+2) together with the class number hK of K divisible by 2l · 3. Then, in this short note, by following the method in [3], we prove that N3,l (X) ≫ X 7/8 for all large enough X’s.

References