The Primes are Everywhere, but Nowhere…

Oleschko, Klaudia; Khrennikov, Andrei; Oleshko, Beatriz F.; Parrot, Jean-François

doi:10.1007/978-3-319-55612-3_7

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Introduction and Main Results1

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2023

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“…For a discussion of the prime numbers' role in a variety of theoretical and practical applications, we suggest [2].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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The average value of a certain number-theoretic function over the primes

Rubin¹

2023

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We consider functions $F:\mathbb{Z}_{\geq 0}\rightarrow\mathbb{Z}_{\geq 0}$ for which there exists a positive integer $n$ such that two conditions hold: $F(p)$ divides $n$ for every prime $p$, and for each divisor $d$ of $n$ and every prime $p$, we have that $d$ divides $F(p)$ iff $d$ divides $F(p \mod d)$. Following an approach of Khrennikov and Nilsson, we employ the prime number theorem for arithmetic progressions to derive an expression for the average value of such an $F$ over all primes $p$, recovering a theorem of these authors as a special case. As an application, we compute the average number of $r$-periodic points of a multivariate power map defined on a product $Z_{f_1(p)}\times\cdots\times Z_{f_m(p)}$ of cyclic groups, where $f_i(t)$ is a polynomial.

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“…For a discussion of the prime numbers' role in a variety of theoretical and practical applications, we suggest [2].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Example 8. Consider the map f : Z 3 × Z 4 given by f (x 1 , x 2 ) = (x 2 1 , x 3 2 ). Here, we can take p = 2, f 1 = x + 1, f 2 = x 2 , n 1 = 2, and n 2 = 3.…”

mentioning

confidence: 99%

The average value of a certain number-theoretic function over the primes

Rubin¹

2023

NNTDM

View full text Add to dashboard Cite

show abstract

The Primes are Everywhere, but Nowhere…

Cited by 1 publication

References 22 publications

The average value of a certain number-theoretic function over the primes

The average value of a certain number-theoretic function over the primes

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