The Ramanujan sum and Chebotarev densities

Wang, Biao

doi:10.1007/s11139-020-00269-8

Cited by 8 publications

(8 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, Kural et al [12] generalized all these results to natural densities of sets of primes. The second author of this article showed the analogues of these results over Q for some arithmetic functions other than µ in [19,21,20]. In this paper, we will show the analogue of Kural et al's result over global function fields.Let p be a prime and let q be a power of prime p. Let F q be a finite field of q elements.…”

mentioning

confidence: 50%

Analogues of Alladi's formula over global function fields

Duan,

Wang,

2020

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we show an analogue of Kural et al.'s result on Alladi's formula for global function fields. Explicitly, we show that for a global function field K, if a set S of prime divisors has a natural density δ(S) within prime divisors, thenwhere µ(D) is the Möbius function on divisors and D(K, S) is the set of all effective distinguishable divisors whose smallest prime factors are in S. As applications, we get the analogue of Dawsey's and Sweeting and Woo's results to the Chebotarev Density Theorem for function fields, and the analogue of Alladi's result to the Prime Polynomial Theorem for arithmetic progressions. We also display a connection between the Möbius function and the Fourier coefficients of modular form associated to elliptic curves. The proof of our main theorem is similar to the approach in Kural et al.'s article.where ϕ is Euler's totient function.Alladi's formula (3) shows a relationship between the Möbius function µ(n) and the density of primes in arithmetic progressions. In 2017, Dawsey [6] first generalized (3) to the setting of Chebotarev densities for finite Galois extensions of Q. Then Sweeting and Woo [17] generalized Dawsey's result to number fields. Recently, Kural et al. [12] generalized all these results to natural densities of sets of primes. The second author of this article showed the analogues of these results over Q for some arithmetic functions other than µ in [19,21,20]. In this paper, we will show the analogue of Kural et al.'s result over global function fields.Let p be a prime and let q be a power of prime p. Let F q be a finite field of q elements. Take K/F p (x) to be a finite extension with constant field F q , which is called a global function field (or simply a function field). A prime divisor (or simply a prime) in K is defined to be a discrete valuation ring R P with maximal ideal P such that F q ⊂ R P and the quotient field of R P is K. The norm of P , denote by |P |, is defined to be the size of the residue field κ P of R P , i.e. |P | = #(R P /P ) = #κ P , which is a power q deg P of the cardinality of the ground field F q . Here the exponent deg P is called the degree of P .

show abstract

mentioning

confidence: 50%

Analogues of Alladi's formula over global function fields

Duan,

Wang,

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…For the arithmetic functions other than µ, we [13,14] showed the analogues of Alladi's and Dawsey's results with respect to the Liouville function and the Ramanujan sum. In this note, we will unify these two analogues by showing the following analogue of formula (4) with respect to the Dirichlet convolutions involving the Möbius function.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 56%

“…For the proof of Theorem 1.1, we follow the approach of proving Theorem 1 in [14]. Then we apply Theorem 1.1 to show (7).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Analogues of Alladi's formula

Wang

2020

Preprint

Self Cite

View full text Add to dashboard Cite

In this note, we mainly show the analogue of one of Alladi's formulas over Q with respect to the Dirichlet convolutions involving the Möbius function µ(n), which is related to the natural densities of sets of primes by recent work of Dawsey, Sweeting and Woo, and Kural et al. This would give us several new analogues. In particular, we get that if (k, ℓ) = 1, thenwhere p(n) is the smallest prime divisor of n, and ϕ(n) is Euler's totient function. This refines one of Hardy's formulas in 1921. At the end, we give some examples for the ϕ(n) replaced by functions "near n", which include the sum-of-divisors function.

show abstract

“…Later, Kural et al [17] generalized all these results to natural densities of sets of prime ideals of number field K. Let P be the set of prime ideals p ⊆ O K and we say that a subset S ⊆ P with natural density δ(S) if the following limit exists: A recent work [8] of Wang with the first and third authors of this article showed the analogue of Kural et al's result over global function fields. In the other direction, Wang [25,26,27] showed the analogues of these results over Q for some arithmetic functions other than µ. More precisely, Wang [27] proved that if an arithmetic function a : N → C satisfies a(1) = 1 and…”

Section: Introduction and Main Resultsmentioning

confidence: 96%

Generalizations of Alladi's formula for arithmetical semigroups

Duan,

Ma,

2021

Preprint

View full text Add to dashboard Cite

show abstract

The Ramanujan sum and Chebotarev densities

Cited by 8 publications

References 10 publications

Analogues of Alladi's formula over global function fields

Analogues of Alladi's formula over global function fields

Analogues of Alladi's formula

Generalizations of Alladi's formula for arithmetical semigroups

Contact Info

Product

Resources

About