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On the Least Prime Represented by a Positive-Definite Binary Quadratic Form.

Record Type:	Electronic resources : Monograph/item
Title/Author:	On the Least Prime Represented by a Positive-Definite Binary Quadratic Form./
Author:	Gaudet, Louis Mayer.
Description:	1 online resource (112 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 84-10, Section: B.
Contained By:	Dissertations Abstracts International84-10B.
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=30317899click for full text (PQDT)
ISBN:	9798379439330

On the Least Prime Represented by a Positive-Definite Binary Quadratic Form.
Gaudet, Louis Mayer.

On the Least Prime Represented by a Positive-Definite Binary Quadratic Form. - 1 online resource (112 pages)

Source: Dissertations Abstracts International, Volume: 84-10, Section: B.

Thesis (Ph.D.)--Rutgers The State University of New Jersey, School of Graduate Studies, 2023.

Includes bibliographical references

In this work, we address the question, "how large is the least prime p of the form p = x 2 + Dy2 relative to D?" More generally, we study the least prime represented by a positive-definite binary quadratic form of prime discriminant D, and we prove a Linnik-type theorem: the least such prime is bounded by a constant times DL for some large but explicit constant L > 0. While such a result has been established before, our methods are significantly different and based on sieve theoretic machinery. In particular, other proofs of this result require deeper input from the zeros of class group L-functions, namely a log-free zero-density estimate and a quantitative form of the Deuring-Heilbronn phenomenon. By comparison, the only input we require from the zeros of these L-functions is a zero-free region of classical type. Along the way, we establish a couple of results that may be of independent interest: (1) a large sieve-type inequality for class group characters over almost-primes, and (2) an approximation result for values of L-functions by partial Euler products whose length is comparable (in the log scale) to the conductor of the L-function.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2023

Mode of access: World Wide Web

ISBN: 9798379439330Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

L-functionIndex Terms--Genre/Form:

542853
Electronic books.

On the Least Prime Represented by a Positive-Definite Binary Quadratic Form.
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245 1 0 $a On the Least Prime Represented by a Positive-Definite Binary Quadratic Form.
264 0 $c 2023
300 $a 1 online resource (112 pages)
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500 $a Source: Dissertations Abstracts International, Volume: 84-10, Section: B.
500 $a Advisor: Iwaniec, Henryk.
502 $a Thesis (Ph.D.)--Rutgers The State University of New Jersey, School of Graduate Studies, 2023.
504 $a Includes bibliographical references
520 $a In this work, we address the question, "how large is the least prime p of the form p = x 2 + Dy2 relative to D?" More generally, we study the least prime represented by a positive-definite binary quadratic form of prime discriminant D, and we prove a Linnik-type theorem: the least such prime is bounded by a constant times DL for some large but explicit constant L > 0. While such a result has been established before, our methods are significantly different and based on sieve theoretic machinery. In particular, other proofs of this result require deeper input from the zeros of class group L-functions, namely a log-free zero-density estimate and a quantitative form of the Deuring-Heilbronn phenomenon. By comparison, the only input we require from the zeros of these L-functions is a zero-free region of classical type. Along the way, we establish a couple of results that may be of independent interest: (1) a large sieve-type inequality for class group characters over almost-primes, and (2) an approximation result for values of L-functions by partial Euler products whose length is comparable (in the log scale) to the conductor of the L-function.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2023
538 $a Mode of access: World Wide Web
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
653 $a L-function
653 $a Prime
653 $a Quadratic form
653 $a Sieve
653 $a Heilbronn phenomenon
655 7 $a Electronic books. $2 lcsh $3 542853
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690 $a 0642
710 2 $a ProQuest Information and Learning Co. $3 783688
710 2 $a Rutgers The State University of New Jersey, School of Graduate Studies. $b Mathematics. $3 3698280
773 0 $t Dissertations Abstracts International $g 84-10B.
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=30317899 $z click for full text (PQDT)