The Prime Number Theorem with an Error Term

2020 ◽  
pp. 237-250
Author(s):  
Tarlok Nath Shorey
Keyword(s):  
Prime Number ◽  
Error Term ◽  
Prime Number Theorem

scholarly journals Integers free of small prime factors in arithmetic progressions

2000 ◽  
Vol 157 ◽  
pp. 103-127 ◽  
Author(s):  
Ti Zuo Xuan
Keyword(s):  
Prime Number ◽  
Short Range ◽  
Error Term ◽  
Real Zero ◽  
Prime Number Theorem ◽  
Arithmetic Progressions ◽  
Real Character ◽  
Dickman Function ◽  
Prime Factors ◽  
Positive Integers

For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfying n ≡ a (mod q). By the fundamental lemma of sieve, it follows that for (a,q) = 1, Φ(x,y;a,q) = φ(q)-1, Φ(x, y){1 + O(exp(-u(log u- log2 3u- 2))) + (u = log x log y) holds uniformly in a wider ranges of x, y and q.Let χ be any character to the modulus q, and L(s, χ) be the corresponding L-function. Let be a (‘exceptional’) real character to the modulus q for which L(s, ) have a (‘exceptional’) real zero satisfying > 1 - c0/log q. In the paper, we prove that in a slightly short range of q the above first error term can be replaced by where ρ(u) is Dickman function, and ρ′(u) = dρ(u)/du.The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that 1 < q < (log q)A.

scholarly journals Updating the error term in the prime number theorem

2015 ◽  
Vol 39 (2) ◽  
pp. 225-234 ◽  
Author(s):  
Tim Trudgian
Keyword(s):  
Prime Number ◽  
Error Term ◽  
Prime Number Theorem

scholarly journals The error term in the prime number theorem

2020 ◽  
Vol 90 (328) ◽  
pp. 871-881 ◽  
Author(s):  
David J. Platt ◽  
Timothy S. Trudgian
Keyword(s):  
Prime Number ◽  
Error Term ◽  
Prime Number Theorem

scholarly journals On the Wintner-Ingham-Segal summability method

2019 ◽  
Author(s):  
S Kanemitsu ◽  
T Kuzumaki ◽  
Y Tanigawa
Keyword(s):  
Prime Number ◽  
Error Term ◽  
Prime Number Theorem ◽  
Summability Method ◽  
Stieltjes Integration ◽  
International Audience

International audience The aim of this note is to establish a subclass of $\mathcal{F}$ considered by Segal if functions for which the Ingham-Wintner summability implies $\mathcal{F}$-summability as wide as possible. The subclass is subject to the estimate for the error term of the prime number theorem. We shall make good use of Stieltjes integration which elucidates previous results obtained by Segal.

scholarly journals On the Prime Geodesic Theorem for SL4

2020 ◽  
Vol 14 ◽  
Keyword(s):  
Prime Number ◽  
Error Term ◽  
Symmetric Spaces ◽  
Elementary Proof ◽  
Real Variable ◽  
Prime Number Theorem ◽  
Class Numbers ◽  
Prime Geodesic Theorem ◽  
Real Quadratic ◽  
Number Of Authors

In 1949, A. Selberg discovered a real variable (an elementary) proof of the prime number theorem. A number of authors have adapted Selberg’s method to achieve quite a good corresponding error term. The Riemann hypothesis has never been proved or disproved however. Any generalization of the prime number theorem to the more general situations is known in literature as a prime geodesic theorem. In this paper we derive yet another proof of the prime geodesic theorem for compact symmetric spaces formed as quotients of the Lie group SL4 (R). While the first known proof in this setting applies contour integration over square boundaries, our proof relies on an application of modified circular boundaries. Recently, A. Deitmar and M. Pavey applied such prime geodesic theorem to derive an asymptotic formula for class numbers of orders in totally complex quartic fields with no real quadratic subfields.

scholarly journals Oscillations of the error term in the prime number theorem

2018 ◽  
Vol 156 (2) ◽  
pp. 303-308
Author(s):  
J.-C. Schlage-Puchta
Keyword(s):  
Prime Number ◽  
Error Term ◽  
Prime Number Theorem
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