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 -