1.7. Notes 29

Chebyshev was the first mathematician to actually prove any results on the

distribution of the primes of the kind envisioned by Gauss and Legendre. He

published two papers on this topic, in 1848 and in 1850. In the first paper [Che48]

Chebyshev shows that the limit

lim

s→1+

∞

n=2

π(n + 1) − π(n) −

1

log(n)

logm(n)

ns

exists and is finite for any choice of the integer m. This is a weak formulation

of the statement that the density of the primes is 1/ log(x). To prove this result,

he uses the Euler product formula that we shall consider in Chapter 3. He then

deduces that for any ε 0 and any integer m each of the inequalities π(x)

li(x) − εx/

logm(x)

and π(x) li(x) + εx/

logm(x)

has arbitrarily large solutions x.

Now it is immediately clear that if the ratio π(x)/li(x) tends to a limit, that limit

must be 1, so that then the relative error in the approximation of Gauss tends to zero

as x → ∞. Chebyshev goes on to conclude that if we put π(x) = x/(log(x) − A(x))

and A(x) has a limit as x → +∞, then the limit must be 1. So if there is an

asymptotically best approximation of the form π(x) ≈ x/(log(x) − A) at all, that

must be the approximation with A = 1 rather than with A = 1.08366 as proposed by

Legendre. In his 1850 paper [Che50] Chebyshev proved an unconditional estimate

that is equivalent to 0.921·li(x) ≤ π(x) ≤ 1.106·li(x) for all x suﬃciently large, by a

rather more elaborate version of the method that we used to prove Proposition 1.16.

Even better upper and lower bounds were obtained by J. J. Sylvester [Syl81] using

Chebyshev’s method. Long after this work Diamond and Erd¨ os [DE80] showed by

means of the Prime Number Theorem that with suﬃcient calculation the method

will yield estimates c1·li(x) ≤ π(x) ≤ c2·li(x) with c1 and c2 arbitrarily close to 1.

The formula relating ψ and T was discovered independently of Chebyshev by

A. de Polignac [dP51].

Chebyshev’s proof of Bertrand’s postulate in his 1850 paper [Che50] inaugu-

rated the study of the local distribution of primes. Any nontrivial bound for the

error term in the Prime Number Theorem implies existence of primes in short inter-

vals. How short one can take the intervals by this approach depends on the quality

of the bound on the error term in the PNT. However, in 1930 G. Hoheisel [Hoh30]

by means of a different, analytic kind of argument succeeded in proving that the

interval (x −

xθ,x]

contains a prime for all x suﬃciently large, with the exponent

θ = 1 − 1/33000. Even today, a bound for the error term in the Prime Number

Theorem strong enough to allow this conclusion is not known. The exponent θ

was gradually reduced over the years, by H. A. Heilbronn [Hei33] (θ = 0.996 in

1933), N. G. Chudakov [Chu36] (θ = 3/4 + ε in 1936), A. E. Ingham [Ing37]

(θ = 5/8 + ε in 1937), H. L. Montgomery [Mon71] (θ = 3/5 + ε in 1971), M. N.

Huxley [Hux72] (θ = 7/12+ε in 1972), H. Iwaniec and M. Jutila [IJ79] (θ = 13/23

in 1979), Heath-Brown and Iwaniec [HBI79] (θ = 11/20 in 1979), J. Pintz [Pin84]

(θ = 17/31−c for some small computable c 0 in 1984), Iwaniec and Pintz [IP84]

(θ = 23/42 in 1984), C. J. Mozzochi [Moz86] (θ = 11/20 − 1/384 in 1986), S. T.

Lou and Q. Yao [LY92, LY93] (θ = 6/11 in 1992 and θ = 6/11 in 1993), R. C.

Baker and G. Harman [BH96] (θ = .535 in 1996), and Baker, Harman and Pintz

[BHP01] (θ = 0.525 in 2001.) There are heuristic arguments in favor of stronger

conclusions. See page 422 of Multiplicative Number Theory I. Classical Theory by