Watt's mean value theorem and Carmichael numbers

被引：21

作者：

Harman, Glyn ^{[1
]}

机构：

[1] Univ London, Royal Holloway & Bedford New Coll, Dept Math, Egham TW20 0EX, Surrey, England

来源：

INTERNATIONAL JOURNAL OF NUMBER THEORY | 2008年 / 4卷 / 02期

基金：

英国工程与自然科学研究理事会;

关键词：

mean value theorems; Dirichlet characters; L-functions; primes in arithmetic progressions; Carmichael numbers; sieve methods; Buchstab's identity;

D O I：

10.1142/S1793042108001316

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

It is shown that Watt's new mean value theorem on sums of character sums can be included in the method described in the author's recent work [6] to show that the number of Carmichael numbers up to x exceeds x(1/3) for all large x. This is done by comparing the application of Watt's original version of his mean value theorem [8] to the problem of primes in short intervals [3] with the problem of finding "small" primes in an arithmetic progression.

引用

页码：241 / 248

页数：8

共 9 条

[1]

Baker RC, 1998, ACTA ARITH, V83, P331

[2] The difference between consecutive primes, II [J].

Baker, RC ;

Harman, G ;

Pintz, J .

PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 2001, 83 :532-562

[3]

Chernick J., 1935, B AM MATH SOC, V45, P269

[4]

Granville A, 2002, MATH COMPUT, V71, P883, DOI 10.1090/S0025-5718-01-01355-2

[5] On the number of Carmichael numbers up to x [J].

Harman, G .

BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 2005, 37 :641-650

[6]

Pinch R. G. E., CARMICHAEL NUMBERS 1

[7] KLOOSTERMAN SUMS AND A MEAN-VALUE FOR DIRICHLET POLYNOMIALS [J].

WATT, N .

JOURNAL OF NUMBER THEORY, 1995, 53 (01) :179-210

[8] Bounds for a mean value of character sums [J].

Watt, Nigel .

INTERNATIONAL JOURNAL OF NUMBER THEORY, 2008, 4 (02) :249-293

[9]

[No title captured]

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