What is the number of rolls of fair six-sided dice until the first time the total sum of all rolls is a prime? We compute the expectation and the variance of this random variable up to an additive ...error of less than
. This is a solution to a puzzle suggested by DasGupta in the Bulletin of the Institute of Mathematical Statistics, where the published solution is incomplete. The proof is simple, combining a basic dynamic programming algorithm with a quick Matlab computation and basic facts about the distribution of primes.
Let Λ(n) be the von Mangoldt function, and let t be the integral part of real number t. In this note, we prove that for any ɛ>0 the asymptotic formula ∑n≤xΛ(xn)=x∑d≥1Λ(d)d(d+1)+Oɛ(x9/19+ɛ)(x→∞) ...holds. This improves a recent result of Bordellès, which requires 97203 in place of 919.
In 2020, Bergelson and Richter gave a dynamical generalization of the classical Prime Number Theorem, which has been generalized by Loyd in a disjoint form with the Erdős-Kac Theorem. These ...generalizations reveal the rich ergodic properties of the number of prime divisors of integers. In this article, we show a new generalization of Bergelson and Richter's Theorem in a disjoint form with the distribution of the largest prime factors of integers. Then following Bergelson and Richter's techniques, we will show the analogues of all of these results for the arithmetic semigroups arising from finite fields as well.
Let n and k be positive integers with
$n\ge k+1$
and let
$\{a_i\}_{i=1}^n$
be a strictly increasing sequence of positive integers. Let
$S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$
.... In 1978, Borwein ‘A sum of reciprocals of least common multiples’, Canad. Math. Bull. 20 (1978), 117–118 confirmed a conjecture of Erdős by showing that
$S_{n,1}\le 1-{1}/{2^{n-1}}$
. Hong ‘A sharp upper bound for the sum of reciprocals of least common multiples’, Acta Math. Hungar. 160 (2020), 360–375 improved Borwein’s upper bound to
$S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$
and derived optimal upper bounds for
$S_{n,2}$
and
$S_{n,3}$
. In this paper, we present a sharp upper bound for
$S_{n,4}$
and characterise the sequences
$\{a_i\}_{i=1}^n$
for which the upper bound is attained.
Analogues of Alladi's formula Wang, Biao
Journal of number theory,
April 2021, 2021-04-00, Volume:
221
Journal Article
Peer reviewed
Open access
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, then−∑n⩾2p(n)≡ℓ(modk)μ(n)φ(n)=1φ(k), where 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.
In this paper, we show an analogue of Kural, McDonald and Sah'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, then−limn→∞∑1≤degD≤nD∈D(K,S)μ(D)|D|=δ(S), where 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.
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