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Open AccessFeature PaperArticle

On the Arithmetic Average of the First n Primes

by Matt Visser

Mathematics 2025, 13(14), 2279; https://doi.org/10.3390/math13142279 - 15 Jul 2025

Viewed by 459

The arithmetic average of the first n primes,

{\bar{p}}_{n} = \frac{1}{n} \sum_{i = 1}^{n} p_{i}

, exhibits very many interesting and subtle properties. Since the transformation from

p_{n} \to {\bar{p}}_{n}

is extremely easy to [...] Read more.

The arithmetic average of the first n primes,

{\bar{p}}_{n} = \frac{1}{n} \sum_{i = 1}^{n} p_{i}

, exhibits very many interesting and subtle properties. Since the transformation from

p_{n} \to {\bar{p}}_{n}

is extremely easy to invert,

p_{n} = n {\bar{p}}_{n} - (n - 1) {\bar{p}}_{n - 1}

, it is clear that these two sequences

p_{n} ⟷ {\bar{p}}_{n}

must ultimately carry exactly the same information. But the averaged sequence

{\bar{p}}_{n}

, while very closely correlated with the primes, (

{\bar{p}}_{n} \sim \frac{1}{2} p_{n}

), is much “smoother” and much better behaved. Using extensions of various standard results, I shall demonstrate that the prime-averaged sequence

{\bar{p}}_{n}

satisfies prime-averaged analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Fourges, Firoozbakht, Nicholson, and Farhadian conjectures. (So these prime-averaged analogues are not conjectures; they are theorems). The crucial key to enabling this pleasant behaviour is the “smoothing” process inherent in averaging. While the asymptotic behaviour of the two sequences is very closely correlated, the local fluctuations are quite different. Full article

28 pages, 2061 KB

Open AccessArticle

Predicting Maximal Gaps in Sets of Primes

by Alexei Kourbatov and Marek Wolf

Mathematics 2019, 7(5), 400; https://doi.org/10.3390/math7050400 - 4 May 2019

Cited by 5 | Viewed by 13176

Abstract

Let

q > r \geq 1

be coprime integers. Let

P_{c} = P_{c} (q, r, H)

be an increasing sequence of primes p satisfying two conditions: (i)

p \equiv r

(mod q) and (ii) p starts [...] Read more.

Let

q > r \geq 1

be coprime integers. Let

P_{c} = P_{c} (q, r, H)

be an increasing sequence of primes p satisfying two conditions: (i)

p \equiv r

(mod q) and (ii) p starts a prime k-tuple with a given pattern

H

. Let

π_{c} (x)

be the number of primes in

P_{c}

not exceeding x. We heuristically derive formulas predicting the growth trend of the maximal gap

G_{c} (x) = {max}_{p^{'} \leq x} (p^{'} - p)

between successive primes

p, p^{'} \in P_{c}

. Extensive computations for primes up to

10^{14}

show that a simple trend formula

G_{c} (x) \sim \frac{x}{π_{c} (x)} \cdot (\log π_{c} (x) + O_{k} (1))

works well for maximal gaps between initial primes of k-tuples with

k \geq 2

(e.g., twin primes, prime triplets, etc.) in residue class r (mod q). For

k = 1

, however, a more sophisticated formula

G_{c} (x) \sim \frac{x}{π_{c} (x)} \cdot (\log \frac{π_{c}^{2} (x)}{x} + O (\log q))

gives a better prediction of maximal gap sizes. The latter includes the important special case of maximal gaps in the sequence of all primes (

k = 1

q = 2

r = 1

). The distribution of appropriately rescaled maximal gaps

G_{c} (x)

is close to the Gumbel extreme value distribution. Computations suggest that almost all maximal gaps satisfy a generalized strong form of Cramér’s conjecture. We also conjecture that the number of maximal gaps between primes in

P_{c}

below x is

O_{k} (\log x)

. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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12 pages, 301 KB

Open AccessArticle

Large Deviation Results and Applications to the Generalized Cramér Model

by Rita Giuliano and Claudio Macci

Mathematics 2018, 6(4), 49; https://doi.org/10.3390/math6040049 - 2 Apr 2018

Viewed by 3172

Abstract

In this paper, we prove large deviation results for some sequences of weighted sums of random variables. These sequences have applications to the probabilistic generalized Cramér model for products of primes in arithmetic progressions; they could lead to new conjectures concerning the (non-random) [...] Read more.

(This article belongs to the Special Issue Stochastic Processes with Applications)

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