TY - JOUR

T1 - The least prime number represented by a binary quadratic form

AU - Sardari, Naser Talebizadeh

N1 - Funding Information:
Acknowledgments. I would like to thank the anonymous referee for the careful reading of the paper, his comments, and pointing out some inaccuracies in the earlier version. I would like to thank Professor Heath-Brown for several insightful and inspiring conversations during the Spring 2017 at MSRI. Professors Radziwill and Soundararajan kindly outlined the proof of Lemma 2.7. Furthermore, I would like to thank Professor Rainer Schulze-Pillot for his comments regarding the Siegel mass formula. I am also grateful to Professors Peter Sarnak, Simon Marshall, Asif Ali Zaman, Masoud Zargar for their comments and encouragement. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1902185 and Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester. The author is grateful to Max Planck Institute for Mathematics in Bonn for its hospitality and financial support.
Publisher Copyright:
© 2021 European Mathematical Society

PY - 2021

Y1 - 2021

N2 - Let D < 0 be a fundamental discriminant and h(D) be the class number of Q(√D). Let R(X, D) be the number of classes of the binary quadratic forms of discriminant D which represent a prime number in the interval [X, 2X]. Moreover, assume that πD(X) is the number of primes which split in Q(√D) with norm in the interval [X, 2X]. We prove that ( π π(X) D(X) )2 R(X, D) (1 + π(X) h(D) ) , h(D) where π(X) is the number of primes in the interval [X, 2X] and the implicit constant in is independent of D and X.

AB - Let D < 0 be a fundamental discriminant and h(D) be the class number of Q(√D). Let R(X, D) be the number of classes of the binary quadratic forms of discriminant D which represent a prime number in the interval [X, 2X]. Moreover, assume that πD(X) is the number of primes which split in Q(√D) with norm in the interval [X, 2X]. We prove that ( π π(X) D(X) )2 R(X, D) (1 + π(X) h(D) ) , h(D) where π(X) is the number of primes in the interval [X, 2X] and the implicit constant in is independent of D and X.

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U2 - 10.4171/JEMS/1031

DO - 10.4171/JEMS/1031

M3 - Article

AN - SCOPUS:85103587547

SN - 1435-9855

VL - 23

SP - 1161

EP - 1223

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 4

ER -