DIETRICH BURDE. do not hold for all x 30, contrary to some references. These estimates on π(x) came up recently in papers on algebraic number theory.

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1 A REMARK ON AN INEQUALITY FOR THE PRIME COUNTING FUNCTION DIETRICH BURDE log() Abstract. We note that the inequalities 0.9 < π() < 1.11 log() do not hold for all 30, contrary to some references. These estimates on π() came up recently in papers on algebraic number theory. 1. Chebyshev s estimates for π() Let π() denote the number of primes not greater than, i.e., π() = p 1. One of the first works on the function π() is due to Chebyshev. He proved (see []) in 185 the following eplicit inequalities for π(), holding for all 0 with some 0 sufficiently large: log() < π() < c log(), = log( 1/ 3 1/3 5 1/5 30 1/30 ) , c = This can be found in many books on analytic number theory (see for eample [1], [3], [11] and [14]). But it seems that this result is sometimes cited incorrectly: it is claimed that the estimates are valid for all 30. For eample, in [6], page 1 we read that log() < π() < c log(), 30. But a quick numerical computation shows that this is wrong. To give an eample, take = 100. Then we have π() = 5 and c < 5. log() Actually, the inequality is far from true for small. We have the following result: Date: October 9,

2 D. BURDE Theorem 1.1. Let c be Chebyshev s constant. Then the inequality π() < c log() is true for all For = it is false. Proof. In [10] it is shown that π() < log() 1.11, 4. The RHS is less or equal to c / log() if and only if ( ) 1.11 c ep c 1 This shows the claim for Since / log() is a monotonously increasing function it is enough to check the claimed estimate for intergers in the intervall [96098, 11006] by computer. For = we have π(96097) = 960 and c / log() The incorrect inequality was also used in a former version of Khare s proof of Serre s modularity conjecture for the level one case, see [8], [9]. Let F be a finite field of characteristic p. The conjecture stated that an odd, irreducible Galois representation ρ: Gal(Q/Q) GL (F) which is unramified outside p is associated to a modular form on SL (Z). Khare s proof is an elaborate induction on p. Starting with a p for which the conjecture is known one wants to prove the conjecture for a larger prime P. Kahre s arguments do only work if P and p are not Fermat primes, and if P p a for certain values a > 1, close to 1. At this point Khare used the incorrect estimate on π(), as eplained above. Fortunately the proof easily could be repaired by using better estimates on π() provided by Rosser and Schoenfeld [1], and Dusart [4]. Indeed, P. Dusart proved inequalities for π() which are much better than Chebyshev s estimates. He verifies this for smaller numerically. Nevertheless he claims in his thesis [5], that Chebyshev gave the following inequality 0.9 log() < π() < 1.11 log(), 30, which is equally wrong. The question is: where lies the origin for this error? Chebyshev himself proved inequalities in [] with his constants and c = 6c 5 1 indeed for all 30, but for inequalities involving ψ() = n Λ(n) instead of π(). His estimates concerning ψ() seem to be correct for all

3 30. For eample, he shows by elementary means that, for all 30, 3 ψ() < ψ() > 5 log() log(6) log () + 5 log() + 1, 4 To derive from this inequalities on π() for 30, we have to estimate ψ() = p [ ] log() log(p). log(p) Using the estimates [y] y < [y] + 1 [y] for y 1 we obtain ψ() π() log() ψ(),. On the RHS we cannot do easily much better than ψ(). Hence we obtain log() < π() < c log(), 30. On the other hand we know that π() = ψ() ( ) log() + O log,, () so that we obtain, as tends to infinity, ( + o(1)) log() π() (c + o(1)) log(). Chebyshev used these estimates to prove Bertrand s postulate: each interval (n, n] for n 1 contains at least one prime. Moreover his results were a first step towards the proof of the prime number theorem.. Other estimates for π() There are many interesting inequalities on the function π(). Let us first consider inequalities of the form A log() < π() < B log()

4 4 D. BURDE for all 0, where 0 depends on the constant A 1 and respectively on B > 1. On the LHS we can choose A equal to 1, if 17. In fact, we have [5] < π(), 17. log() Note that for = we have / log() , but π() = 6. Consider the RHS of the above inequalities: if we want to hold such inequalities on π() for all 0 with a smaller 0, we need to enlarge the constant B. Conversely, if we need this inequality for smaller B, we have to enlarge 0. The prime number theorem ensures that we can choose B as close to 1 as we want, provided 0 is sufficiently large. The following result of Dusart [4] enables us to derive adjusted versions for the above inequalities: Theorem.1 (Dusart). For real we have the following sharp bounds: π() π() ( log() log() log() log () ( log() +.51 log () ), 399, ), One can derive, for eample, the following inequalities. π() < log(), 84860, π() < log(), 17. Among other inequalities on π() we mention the following ones: log() m < π() < log() M for all 0 with real constants m and M. They have been studied by various authors. A good reference is the article [10]. There it is shown, for eample, that π() > log() 8, 399, 9 π() < log() 1.11, 4. The second inequality can also be used to obtain results on our estimate π() < B, in log() particular for smaller, where the second inequality of Theorem.1 is not valid. However we have ( log() log() +.51 ) log < () log() 1.11, 8516.

5 For > 10 6 and a = we can use [10] π() < log() a. Here the upper bound of Dusart is better only as long as Finally we mention the book [13], providing many references on inequalities on π(), and the recent article [7], where lower and upper bounds for π() of the form n H n c are discussed, where H n = n 3. acknowledgement We are grateful to the referee for drawing our attention to several approimations of π(). We thank J. Sándor for helpful remarks. References [1] T. Apostol: Introduction to analytic number theory. Springer Verlag, fünfte Auflage [] P. L. Chebyshev: Mémoire sur les nombres premiers. Journal de Math. Pures et Appl. 17 (185), [3] K. Chandrasekharan: Introduction to analytic number theory, Moscow [4] P. Dusart: Inégalités eplicites pour ψ(x), θ(x), π(x) et les nombres premiers. C. R. Math. Acad. Sci. Soc. R. Can. 1 (1999), no., [5] P. Dusart: Autour de la fonction qui compte le nombre de nombres premiers. Thése de Doctorat de l Université de Limoges (1998). [6] W. and F. Ellison: Prime numbers. Wiley Interscience [7] M. Hassani: Approimation of π() by Ψ(). J. of Inequ. in Pure and Apl. Math. 7(1) (006), 1-7. [8] C. Khare: On Serre s modularity conjecture for -dimensional mod p representations of Gal(Q/Q) unramified outside p. ArXiv math.nt/ (005). [9] C. Khare: Serre s modularity conjecture: the level one case. To appear in Duke Math. J. [10] L. Panaitopol: Several approimations of π(). Math. Inequal. & Appl. No. 3 (1999), [11] K. Prachar: Primzahlverteilung. Grundlehren der Mathematischen Wissenschaften 91, Springer-Verlag, Berlin-New York, [1] J. B. Rosser, L. Schoenfeld: Sharper bounds for the Chebyshev functions θ() and ψ(). Math. Comp. 9 (1975), [13] J. Sándor, D. S. Mitrinovic, B. Crstici: Handbook of number theory I. Springer, 006. [14] G. Tenenbaum: Introduction to analytic and probabilistic number theory. Cambridge studies in advanced mathematics 46 (1995). address: dietrich.burde@univie.ac.at Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien 5