Explicit bounds on primes

Collecting references: [Dusart, 1998 †Dusart, P. 1998
Autour de la fonction qui compte le nombre de nombres premiers
Ph.D. thesis, Limoges, http://www.unilim.fr/laco/theses/1998/T1998_01.pdf. 173 pp.
],
1. Bounds on primes, without any congruence condition
[Rosser, 1941 †Rosser, J.B. 1941
Explicit bounds for some functions of prime numbers
American Journal of Math., 63, 211--232.
], [Rosser & Schoenfeld, 1962 †Rosser, J.B., & Schoenfeld, L. 1962
Approximate formulas for some functions of prime numbers
Illinois J. Math., 6, 64--94.
], [Rosser & Schoenfeld, 1975 †Rosser, J.B., & Schoenfeld, L. 1975
Sharper bounds for the Chebyshev Functions ϑ(X) and ψ(X)
Math. Comp., 29(129), 243--269.
], [Schoenfeld, 1976 †Schoenfeld, L. 1976
Sharper bounds for the Chebyshev Functions ϑ(X) and ψ(X) II
Math. Comp., 30(134), 337--360.
], [Dusart, 1999a †Dusart, P. 1999a
Inégalités explicites pour ψ(X), θ(X), π(X) et les nombres premiers
C. R. Math. Acad. Sci., Soc. R. Can., 21(2), 53--59.
], [Dusart, 2010 †Dusart, P. 2010
Estimates of some functions over primes without R. H
http://arxiv.org/abs/1002.0442.
],
2. Bounds on the n-th prime
Denote by pn the n-th prime. Thus p1=2,p2=3,p4=5,. The classical form of prime number theorem yields easily pnnlogn. [Rosser, 1938 †Rosser, J.B. 1938
The n-th prime is greater than nlogn
Proc. Lond. Math. Soc., II. Ser., 45, 21--44.
] shows that this equivalence does not oscillate by proving that pn is greater than nlogn for n2. The asymptotic formula for pn can be developped as shown in [Cipolla, 1902 †Cipolla, M. 1902
La determinatzione assintotica dell`nimo numero primo
Matematiche Napoli, 3, 132--166.
]:
pnn(logn+loglogn1+loglogn2logn(lnlnn)26loglogn+112log2n+).
This asymptotic expansion is the inverse of the logarithmic integral Li(x) obtained by series reversion. But [Rosser, 1938 †Rosser, J.B. 1938
The n-th prime is greater than nlogn
Proc. Lond. Math. Soc., II. Ser., 45, 21--44.
] also proved that for every n>1:
n(logn+loglogn10)<pn<n(logn+loglogn+8).
He improves these results in [Rosser, 1941 †Rosser, J.B. 1941
Explicit bounds for some functions of prime numbers
American Journal of Math., 63, 211--232.
] : for every n55,
n(logn+loglogn4)<pn<n(logn+loglogn+2).
This result was subsequently improved by Rosser and Schoenfeld [Rosser & Schoenfeld, 1962 †Rosser, J.B., & Schoenfeld, L. 1962
Approximate formulas for some functions of prime numbers
Illinois J. Math., 6, 64--94.
] in 1962 to
n(logn+loglogn3/2)<pn<n(logn+loglogn1/2),
for n>1 and n>19 respectively. The constants were subsequently reduced by [Robin, 1983a †Robin, G. 1983a
Estimation de la fonction de Tchebychef θ sur le k-ième nombres premiers et grandes valeurs de la fonction ω(n) nombre de diviseurs premiers de n
Acta Arith., 42, 367--389.
]. In particular, the lower bound
n(logn+loglogn1.0072629)<pn
is valid for n>1 and the constant 1.0072629 can be replaced by 1 for pk1011. Then [Massias & Robin, 1996 †Massias, J.-P., & Robin, G. 1996
Bornes effectives pour certaines fonctions concernant les nombres premiers
J. Théor. Nombres Bordeaux, 8(1), 215--242.
] showed that the lower bound constant equals to 1 was admissible for pk<e598 and pk>e1800. Finally, [Dusart, 1999b †Dusart, P. 1999b
The kth prime is greater than k(lnk+lnlnk1) for k2
Math. Comp., 68(225), 411--415.
] showed that
n(lognloglogn1)<pn
for all n>1, and also that
pn<n(logn+loglogn0.9484)
for n>39017 i.e. pn>467473.
3. Bounds on primes in arithmetic progressions
[McCurley, 1984a †McCurley, K.S. 1984a
Explicit estimates for the error term in the prime number theorem for arithmetic progressions
Math. Comp., 42, 265--285.
], [McCurley, 1984b †McCurley, K.S. 1984b
Explicit estimates for θ(x;3,) and ψ(x;3,)
Math. Comp., 42, 287--296.
], [Ramaré & Rumely, 1996 †Ramaré, O., & Rumely, R. 1996
Primes in arithmetic progressions
Math. Comp., 65, 397--425.
], [Dusart, 2002 †Dusart, P. 2002
Estimates for θ(x;k,) for large values of x
Math. Comp., 71(239), 1137--1168.
], [Ramaré, 2002 †Ramaré, O. 2002
Sur un théorème de Mertens
Manuscripta Math., 108, 483--494.
]. Lemma 10 of [Moree, 2004 †Moree, P. 2004
Chebyshev's bias for composite numbers with restricted prime divisors
Math. Comp., 73(245), 425--449.
], section 4 of [Moree & te Riele, 2004 †Moree, P., & te Riele, H.J.J. 2004
The hexagonal versus the square lattice
Math. Comp., 73(245), 451--473.
].
4. Least prime verifying a condition
[Bach & Sorenson, 1996 †Bach, E., & Sorenson, J. 1996
Explicit bounds for primes in residue classes
Math. Comp., 65(216), 1717--1735.
], [Kadiri, 2008 †Kadiri, H. 2008
Short effective intervals containing primes in arithmetic progressions and the seven cube problem
Math. Comp., 77(263), 1733--1748.
],

Last updated on December 27th, 2012, by Pierre Dusart