*On the distribution of the prime numbers*

Note of M. J. E. Littlewood, presented by M. J. Hadamard. In *Mathematical Analysis, Comptes Rendus: Séance* du 22 Jun 1914; p.1869.

(*Attempted translation from the French original by Bhupinder Singh Anand on October 14, 2018. Corrections and/or clarifications—especially where marked (?)—are invited, and welcome.*)

1. M. E. Schmidt has established the existence of a positive number , such that

(1)

and

for monotonically (?) increasing values of .

Also, by Riemann’s Hypothesis on the zeros (roots?) of , we have

(2) .

Between (1) and (2) there is a gap which I now propose to narrow (?) by replacing the inequalities (1) by

(3)

and

Clearly the inequality , generally presumed on empirical grounds, would not hold for all values of .

The inequalities (3) are equivalent to the following:

(4) ,

and ,

Where is Tchebichef’s well-known function. ** I shall show (4) on the assumption that the Riemann Hypothesis holds** (?). In the contrary case (?), we already know more (?).

I define , and denote by

, where

the complex zeros of . We know [footnote 1: We can find formulas equivalent to (4), (5), and (6) in the *Handbuch* by M. Landau, p.365, 387,388] that

(5) ,

(6) .

uniformly for ,

(7) .

(8)

where is the inverse function of

.

2. Lemma. — Let be any positive number, and

.

There are values of such that , and

.

I consider, for example, the second inequality. We can show, first, that

(9) ,

when tends to zero. This follows, essentially, by elementary reasoning, from the formula:

,

as a consequence of (8). We can also infer from (8) the existence of an , such that:

(10) .

I shall now apply a theorem due to Kronecker, which M. H. Bohr recognised as of primary importance to the theory of Dirichlet series. Let , where is entire in the neighbourhood (?) of . There is, by Kronecker’s theorem, a such that:

and

for . I choose ; whence, after (10),

.

Thus

(11) .

However

,

and (11) holds for arbitrary small values of ; which can easily be seen to yield the inequality of the lemma.

3. It suffices therefore, in order to establish the relations in (3), to show that an assumption, such that

(12)

for all and , contradicts the lemma.

From (6) and (7), we can show the existence of a curve where (?) , where is positive, continuous, and increasing, such that

(13?)

for . From (5), (12) and (13), we conclude

uniformly in the region . Suppose

We then have on and to the right of . We also have, by (5), (6), (7) and (8),

uniformly in ; whence we conclude

uniformly in the region . Finally, by an extension of a theorem due to Lindelöf, we have

for and , which contradicts the lemma in paragraph 3.

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