New estimates for some integrals of functions defined over primes

Axler, Christian

doi:10.7169/facm/2049

Cited by 6 publications

(7 citation statements)

References 16 publications

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“…Here and after, we will use implicitly the basic observation that a recurrence sequence (x n ) n≥1 as in (1) is eventually periodic if and only if it is bounded, cf. [5, p. 45].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…. , n d } + 1) then there are no eventually periodic sequences (x n ) n≥1 as in (1). This is the starting point for this work, which motivates the following heuristic: if a function f satisfies (2) "on average," then every sequence (x n ) n≥1 as (1) should be eventually periodic, independently of its starting values.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In the proof of Theorem 3, we will need also the effective version of the third Mertens' theorem given by Rosser and Schoenfeld [16] in 1962 (see also [1,3]). As usual, hereafter we reserve the letter p for primes.…”

Section: Proofsmentioning

confidence: 99%

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On the iterates of shifted Euler's function

Leonetti¹,

Luca²

2023

Preprint

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Let ϕ be the Euler's function and fix an integer k ≥ 0. We show that, for every initial value x 1 ≥ 1, the sequence of positive integers (xn) n≥1 defined by x n+1 = ϕ(xn) + k for all n ≥ 1 is eventually periodic. Similarly, for every initial value x 1 , x 2 ≥ 1, the sequence of positive integers (xn) n≥1 defined by x n+2 = ϕ(x n+1 )+ϕ(xn)+k for all n ≥ 1 is eventually periodic, provided that k is even.

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“…Here and after, we will use implicitly the basic observation that a recurrence sequence (x n ) n≥1 as in (1) is eventually periodic if and only if it is bounded, cf. [5, p. 45].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

On the iterates of shifted Euler's function

Leonetti¹,

Luca²

2023

Preprint

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“…In the proof of Theorem 1.3, we will need also the effective version of the third Mertens' theorem given by Rosser and Schoenfeld [16] in 1962 (see also [1,3]). As usual, hereafter, we reserve the letter p for primes.…”

Section: Proof Since ϕ(N) ≤ N −mentioning

confidence: 99%

On the Iterates of the Shifted Euler’s Function

Leonetti

Luca²

2023

Bull. Aust. Math. Soc.

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Let $\varphi $ be Euler’s function and fix an integer $k\ge 0$ . We show that for every initial value $x_1\ge 1$ , the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+1}=\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic. Similarly, for all initial values $x_1,x_2\ge 1$ , the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+2}=\varphi (x_{n+1})+\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic, provided that k is even.

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“…The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1 2 . It is considered by many to be the most important unsolved problem in pure mathematics.…”

Section: Introductionmentioning

confidence: 99%

Breakthrough Results on Prime Numbers

Vega

2024

Preprint

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Let $\Psi(n) = n \cdot \prod_{q \mid n} \left(1 + \frac{1}{q} \right)$ denote the Dedekind $\Psi$ function where $q \mid n$ means the prime $q$ divides $n$. Define, for $n \geq 3$; the ratio $R(n) = \frac{\Psi(n)}{n \cdot \log \log n}$ where $\log$ is the natural logarithm. Let $N_{n} = 2 \cdot \ldots \cdot q_{n}$ be the primorial of order $n$. We state that if the inequality $R(N_{n+1}) < R(N_{n})$ holds for all primes $q_{n}$ (greater than some threshold), then the Riemann hypothesis is true and the Cram{\'e}r's conjecture is false. In this note, we prove that the previous inequality always holds for all sufficiently large primes.

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New estimates for some integrals of functions defined over primes

Cited by 6 publications

References 16 publications

On the iterates of shifted Euler's function

On the iterates of shifted Euler's function

On the Iterates of the Shifted Euler’s Function

Breakthrough Results on Prime Numbers

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