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 $K$, such that

(1) $\Pi(x) - Li\ x + \frac{1}{2}. Li \sqrt{x} < -K. \frac{\sqrt{x}}{log\ x}$

and $\Pi(x) - Li\ x + \frac{1}{2}. Li \sqrt{x} > K. \frac{\sqrt{x}}{log\ x}$

for monotonically (?) increasing values of $x$.

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

(2) $\Pi(x) - Li\ x = O(\sqrt{x}. log\ x)$.

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

(3) $\Pi(x) - Li\ x < -K. \frac{\sqrt{x}. log log log\ x}{log\ x}$

and $\Pi(x) - Li\ x > K. \frac{\sqrt{x}. log log log\ x}{log\ x}$

Clearly the inequality $\Pi(x) < Li\ x$, generally presumed on empirical grounds, would not hold for all values of $x$.

The inequalities (3) are equivalent to the following:

(4) $\psi (x) - x < -K. \sqrt{x}. log log log\ x$,

and $\psi (x) - x > K. \sqrt{x}. log log log\ x$,

Where $\psi (x)$ 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 $\eta = log\ x$, and denote by

$\frac{1}{2} \pm i\gamma_{1},\ \frac{1}{2} \pm i\gamma_{2},\ \ldots$, where $0 < \gamma_{1} \leq \gamma_{2} \leq \ldots$

the complex zeros of $\zeta(s)$. 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) $\frac{\psi(x) - x}{\sqrt{x}} = -2.\sum_{_{1}}^{^{\infty}}\frac{sin \gamma_{n} \eta}{\gamma_{n}} + O(1)$,

(6) $\frac{\psi(x) - x}{\sqrt{x}} = -2.\sum_{_{\gamma_{n} \leq T}}\frac{sin \gamma_{n} \eta}{\gamma_{n}} + O(1)$.

uniformly for $T < x^{^{2}}$,

(7) $\sum_{_{1}}^{^{\infty}}\frac{sin \gamma_{n} \eta}{\gamma_{n}} = O(\eta^{^{2}})$.

(8) $\gamma_{n} = g(n) + O(1)$

where $t = g(T) \sim \frac{2 \pi T}{log\ T}$ is the inverse function of

$T = \frac{t\ log\ t}{2\pi} - \frac{(1+ log\ 2\pi)t}{2\pi}$.

2. Lemma. — Let $\gamma_{_{0}}$ be any positive number, and

$f(z) = f(\xi + i\gamma) = \sum_{_{1}}^{^{\infty}}\frac{e^{^{-\gamma_{n}(\xi + i\eta)}}}{\gamma_{n}}$.

There are values of $z$ such that $0 < \xi \leq 1,\ \eta \geq \eta_{_{0}}$, and

$-\texttt{I}f(z)= \sum_{_{1}}^{^{\infty}} \frac{sin \gamma_{n} \eta}{\gamma_{n}}\ e^{^{-\gamma_{n}\xi}} < -K. log log\ \eta$.

$-\texttt{I}f(z)= \sum_{_{1}}^{^{\infty}} \frac{sin \gamma_{n} \eta}{\gamma_{n}}\ e^{^{-\gamma_{n}\xi}} > K. log log\ \eta$

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

(9) $-\texttt{I}f(\xi + i \xi) = \sum_{_{1}}^{^{n}}\frac{sin \gamma_{_{n}} \xi}{\gamma_{_{n}}}.e^{^{-\gamma_{_{n}}\xi}} \sim A\ log\frac{1}{\xi}\ A > 0$,

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

$f(\xi + i \xi) = \int_{_{1}}^{^{\infty}}\frac{e^{^{-(\xi + i \xi.g(n))}}}{g(n)}.dn + O \int_{_{1}}^{^{\infty}}\frac{e^{^{-\xi.g(n)}}}{g(n)}.(\xi + \frac{1}{n}).dn$,

as a consequence of (8). We can also infer from (8) the existence of an $a > 0$, such that:

(10) $\sum_{_{\gamma_{_{n}}\xi > a}}\frac{e^{^{-\gamma_{_{n}}\xi}}}{\gamma_{_{n}}} > \frac{1}{4}A.log\frac{1}{\xi}$.

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 $(x) = |x - \{x\}|$, where $\{x\}$ is entire in the neighbourhood (?) of $x$. There is, by Kronecker’s theorem, a $T$ such that:

$\eta_{_{0}} < T < \eta_{_{0}}(\frac{1}{\xi} + 1)^{^{N}}$ and $(\frac{\gamma_{_{n}}.T}{2\pi})<\xi$

for $n \leq N$. I choose $N = \frac{a}{\xi},\ \eta = T + \xi$; whence, after (10),

$|\texttt{I}f(\xi + i \eta) - \texttt{I}f(\xi + i \xi)| \leq \sum_{_{1}}^{^{N}}(\frac{|sin \gamma_{_{n}} \eta - sin \gamma_{_{n}} \xi|}{\gamma_{_{n}}}) + 2.\sum_{_{N+1}}^{^{\infty}}(\frac{e^{^{-\gamma_{_{n}}\xi}}}{\gamma_{_{n}}})$

$< 2\pi N \xi + (\frac{1}{2}A + \varepsilon).log\frac{1}{\xi} < 2\pi a + (\frac{1}{2}A + \varepsilon).log\frac{1}{\xi}$.

Thus

(11) $-\texttt{I}f(\xi + i \eta) > K.log\frac{1}{\xi}$.

However

$\eta < \xi + \eta_{_{0}}(\frac{1}{\xi} + 1)^{^{\frac{a}{\xi}}}$,

and (11) holds for arbitrary small values of $\xi$; 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) $\psi(x) - 1 < \delta. \sqrt{x}.log log log\ x$

for all $\delta > 0$ and $x > x_{_{0}}\delta$, contradicts the lemma.

From (6) and (7), we can show the existence of a curve $C$ where (?) $\xi = \xi(\eta)$, where $\xi(\eta)$ is positive, continuous, and increasing, such that

(13?) $|\sum_{_{1}}^{^{?}}\frac{sin \gamma_{_{n}}\eta}{\gamma_{_{n}}} - \sum_{_{1}}^{^{\infty}}\frac{sin \gamma_{_{n}}\eta}{\gamma_{_{n}}}.e^{^{-\gamma_{_{n}}\xi}}| < K$

for $0 < \xi \leq \xi_{_{n}}$. From (5), (12) and (13), we conclude

$-\texttt{I} f(z) < \delta. log log\ \eta$

$[\eta > \eta_{_{0}}(\delta)]$

uniformly in the region $0 < \xi \leq \xi_{_{n}}$. Suppose

$g(z) = e^{^{i f(z)}}(log\ z)^{^{-2.\delta}}$

We then have $g(z) = O(1)$ on $C$ and to the right of $\xi = 1$. We also have, by (5), (6), (7) and (8),

$|\sum_{_{1}}^{^{n}}\frac{sin \gamma_{_{\nu}}\eta}{\gamma_{_{\nu}}}| = O(\eta^{^{2}})$

uniformly in $n$; whence we conclude

$|\texttt{I}f(z)| = O(\eta^{^{2}}),\ g(z) = e^{^{0\eta^{^{2}}}}$

uniformly in the region $\xi(\eta) \leq \xi \leq 1$. Finally, by an extension of a theorem due to Lindelöf, we have

$g(z) = O(1),\ -\texttt{I} f(z) < 3.\delta.log log\ \eta$

for $0 < \xi \leq 1$ and $\eta_{_{0}}(\delta) < \eta$, which contradicts the lemma in paragraph 3.