The function field abstract prime number theorem

1989 ◽  
Vol 106 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Stephen D. Cohen
Keyword(s):  
Finite Field ◽  
Prime Number ◽  
Function Field ◽  
Prime Number Theorem ◽  
Arithmetical Semigroup ◽  
Enumeration Problems ◽  
Ring Of Polynomials ◽  
Arithmetical Semigroups ◽  
Positive Integers ◽  
Free Commutative Semigroup

For arithmetical semigroups modelled on the positive integers, there is an ‘abstract prime number theorem’ (see, for example, [1]). In order to study enumeration problems in the several arithmetical categories whose prototype instead is the ring of polynomials in an indeterminate over a finite field of order q, Knopfmacher[2, 3] introduced the following modification. An additive arithmetical semigroup G is a free commutative semigroup with an identity, generated by a countable set of ‘primes’ P and admitting an integer-valued degree mapping ∂ with the properties(i) ∂(l) = 0,∂(p) > 0 for p∈P;(ii) ∂(ab) = ∂(a) + ∂(b) for all a, b in G;(iii) the number of elements in G of degree n is finite. (This number will be denoted by G(n).)

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 Asymptotic counting theorems for primitive juggling patterns

2019 ◽  
Vol 15 (05) ◽  
pp. 1037-1050
Author(s):  
Erik R. Tou
Keyword(s):  
Prime Number ◽  
Generating Functions ◽  
Arbitrary Number ◽  
Mathematical Theory ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Theoretical Framework ◽  
Positive Integers ◽  
Infinite Set ◽  
Fixed Period

The mathematics of juggling emerged after the development of siteswap notation in the 1980s. Consequently, much work was done to establish a mathematical theory that describes and enumerates the patterns that a juggler can (or would want to) execute. More recently, mathematicians have provided a broader picture of juggling sequences as an infinite set possessing properties similar to the set of positive integers. This theoretical framework moves beyond the physical possibilities of juggling and instead seeks more general mathematical results, such as an enumeration of juggling patterns with a fixed period and arbitrary number of balls. One problem unresolved until now is the enumeration of primitive juggling sequences, those fundamental juggling patterns that are analogous to the set of prime numbers. By applying analytic techniques to previously-known generating functions, we give asymptotic counting theorems for primitive juggling sequences, much as the prime number theorem gives asymptotic counts for the prime positive integers.

scholarly journals The -parity conjecture for abelian varieties over function fields of characteristic

2014 ◽  
Vol 150 (4) ◽  
pp. 507-522 ◽  
Author(s):  
Fabien Trihan ◽  
Seidai Yasuda
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Prime Number ◽  
Abelian Variety ◽  
Function Field ◽  
Function Fields ◽  
Selmer Group ◽  
Smooth Variety ◽  
Tate Conjecture ◽  
Parity Conjecture

AbstractLet $A/K$ be an abelian variety over a function field of characteristic $p>0$ and let $\ell $ be a prime number ($\ell =p$ allowed). We prove the following: the parity of the corank $r_\ell $ of the $\ell $-discrete Selmer group of $A/K$ coincides with the parity of the order at $s=1$ of the Hasse–Weil $L$-function of $A/K$. We also prove the analogous parity result for pure $\ell $-adic sheaves endowed with a nice pairing and in particular for the congruence Zeta function of a projective smooth variety over a finite field. Finally, we prove that the full Birch and Swinnerton-Dyer conjecture is equivalent to the Artin–Tate conjecture.

scholarly journals Sign changes in the prime number theorem

2021 ◽  
Author(s):  
Thomas Morrill ◽  
Dave Platt ◽  
Tim Trudgian
Keyword(s):  
Prime Number ◽  
Prime Number Theorem ◽  
Sign Changes

scholarly journals The restricted Burnside problem for Moufang loops

2021 ◽  
pp. 1-11
Author(s):  
ALEXANDER GRISHKOV ◽  
LIUDMILA SABININA ◽  
EFIM ZELMANOV
Keyword(s):  
Prime Number ◽  
Burnside Problem ◽  
Moufang Loops ◽  
Restricted Burnside Problem ◽  
Positive Integers

Abstract We prove that for positive integers $m \geq 1, n \geq 1$ and a prime number $p \neq 2,3$ there are finitely many finite m-generated Moufang loops of exponent $p^n$ .

scholarly journals The prime number theorem is PRA-provable

2001 ◽  
Vol 257 (1-2) ◽  
pp. 185-239 ◽  
Author(s):  
Olivier Sudac
Keyword(s):  
Prime Number ◽  
Prime Number Theorem

INDEX-DEPENDENT DIVISORS OF COEFFICIENTS OF MODULAR FORMS

2013 ◽  
Vol 09 (07) ◽  
pp. 1841-1853 ◽  
Author(s):  
B. K. MORIYA ◽  
C. J. SMYTH
Keyword(s):  
Prime Number ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Survey Methods ◽  
Integer Sequences ◽  
Positive Integers

We evaluate [Formula: see text] for a certain family of integer sequences, which include the Fourier coefficients of some modular forms. In particular, we compute [Formula: see text] for all positive integers n for Ramanujan's τ-function. As a consequence, we obtain many congruences — for instance that τ(1000m) is always divisible by 64000. We also determine, for a given prime number p, the set of n for which τ(pn-1) is divisible by n. Further, we give a description of the set {n ∈ ℕ : n divides τ(n)}. We also survey methods for computing τ(n). Finally, we find the least n for which τ(n) is prime, complementing a result of D. H. Lehmer, who found the least n for which |τ(n)| is prime.

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