The n
-th Prime is Greater than n
logn

Rosser, Barkley

doi:10.1112/plms/s2-45.1.21

Cited by 78 publications

(44 citation statements)

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“…The estimate (a) is due to Rosser [2] and the estimates (b) and (c) are due to Dusart [1, p. 54]. From Lemma 2.1 (b) and (c), we obtain Lemma 2.2.…”

Section: Lemmasmentioning

confidence: 70%

On a Conjecture on Ramanujan Primes

Laishram

2010

Int. J. Number Theory

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Abstract. For n ≥ 1, the nth Ramanujan prime is defined to be the smallest positive integer Rn with the property that if) ≥ n where π(ν) is the number of primes not exceeding ν for any ν > 0 and ν ∈ R. In this paper, we prove a conjecture of Sondow on upper bound for Ramanujan primes. An explicit bound of Ramanujan primes is also given. The proof uses explicit bounds of prime π and θ functions due to Dusart.

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“…The estimate (a) is due to Rosser [2] and the estimates (b) and (c) are due to Dusart [1, p. 54]. From Lemma 2.1 (b) and (c), we obtain Lemma 2.2.…”

Section: Lemmasmentioning

confidence: 70%

On a Conjecture on Ramanujan Primes

Laishram

2010

Int. J. Number Theory

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“…Since log is monoton we have to estimate-log vl~) (1-1) from above. Now j=l n According to [12] we have pj>j logj. Therefore…”

Section: Get Vol(g(2gp))=p-3p2 (P-(~pl))=(1-p-2) (L + (~)P-t) -1mentioning

confidence: 98%

On the cuspidal cohomology of the Bianchi modular groups

Rohlfs

1985

Math Z

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“…Let a = \n3y/ln2y, and let 8 > 0. We now show that if y > yxi8), we have for each i such that rt exists (6) rl>J»J+<1-3a>a, since by Rosser [20], P¡ < 2/ In i holds for all / > 1. So we now assume z > exp((ln2 jy)2/6 ln3y).…”

Section: The Controversy Concerning the Growth Rate Of C(x)mentioning

confidence: 99%

The pseudoprimes to 25⋅10⁹

Pomerance¹,

Selfridge²,

Wagstaff³

1980

Math. Comp.

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Abstract.The odd composite n < 25 • 10 such that 2n_1 = 1 (mod n) have been determined and their distribution tabulated. We investigate the properties of three special types of pseudoprimes: Euler pseudoprimes, strong pseudoprimes, and Carmichael numbers. The theoretical upper bound and the heuristic lower bound due to Erdös for the counting function of the Carmichael numbers are both sharpened.Several new quick tests for primality are proposed, including some which combine pseudoprimes with Lucas sequences.

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The n -th Prime is Greater than n logn

Cited by 78 publications

References 0 publications

On a Conjecture on Ramanujan Primes

On a Conjecture on Ramanujan Primes

On the cuspidal cohomology of the Bianchi modular groups

The pseudoprimes to 25⋅10⁹

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