Hardy (1914) — A Proof of Infinitely Many Zeros on the Critical Line

In 1914, G. H. Hardy published a short but influential note showing that the Riemann zeta function \(\zeta(s)\) has infinitely many zeros on the critical line \(\sigma = \tfrac12\). This was the first rigorous result of its kind, and it marked the beginning of the "critical-line program" in analytic number theory.

1. Preliminaries

Earlier work by Bohr and Landau had established that, for any \(\delta > 0\), almost all complex zeros of \(\zeta(s)\) lie in the strip \[ \frac12 - \delta < \sigma < \frac12 + \delta. \] Hardy goes further: not only are the zeros concentrated near the critical line, but infinitely many actually lie on it.

The first step is to use a Mellin inversion formula, due to Cahen: \[ e^{-y} = \frac{1}{2\pi i} \int_{k-i\infty}^{k+i\infty} \Gamma(u)\,y^{-u}\,du, \qquad (\Re y > 0,\, k>0). \] Multiplying by \(n^{2u}\) and summing over \(n\ge1\) produces \[ \sum_{n=1}^\infty e^{-n^2 y} = \frac{1}{2\pi i} \int_{k-i\infty}^{k+i\infty} \Gamma(u)\,y^{-u}\,\zeta(2u)\,du, \qquad (k > \tfrac12). \]

Riemann's functional equation allows this integral to be rewritten in terms of the real entire function \[ \Xi(t) := \xi\!\left(\tfrac12 + it\right), \] which satisfies \(\Xi(t) \in \mathbb{R}\) for real \(t\). After shifting contours and simplifying, Hardy arrives at

\[ \int_0^\infty \frac{(e^{\alpha t} + e^{-\alpha t})\,\Xi(2t)} {\tfrac14 + 4t^2}\,dt = \pi \cos\!\left(\tfrac{\alpha\pi}{4}\right) -\frac{\pi}{2} e^{i\alpha\pi/4} F(q), \tag{1} \]

where \[ F(q) = 1 + 2\sum_{n=1}^\infty q^{n^2}, \qquad q = e^{-\pi e^{4\alpha}}. \] This theta-type series is well behaved as long as \(q\) avoids the positive real axis.

2. Differentiation and limiting behavior

Differentiating equation (1) \(2p\) times with respect to \(\alpha\) yields

\[ \int_0^\infty \frac{(e^{\alpha t} + e^{-\alpha t})\,t^{2p}\,\Xi(2t)} {\tfrac14 + 4t^2}\,dt = \frac{(-1)^p\pi}{4^p} \cos\!\left(\tfrac{\alpha\pi}{4}\right) - \left(\frac{d}{d\alpha}\right)^{2p} \!\left[\frac{\pi}{2}e^{i\alpha\pi/4}F(q)\right]. \tag{2} \]

A key lemma from the theory of Dirichlet series (Bohr–Riesz summability) shows that as \(\alpha \to \tfrac{\pi}{2}\), the parameter \(q = e^{-\pi e^{4\alpha}}\) approaches \(-1\) in a controlled way, and every term \(\left(\tfrac{d}{d\alpha}\right)^{2p}[e^{i\alpha\pi/4}F(q)]\) tends to zero. Thus, equation (2) simplifies, in the limit, to

\[ \int_0^\infty \frac{(e^{\alpha t} + e^{-\alpha t})\,t^{2p}\,\Xi(2t)} {\tfrac14 + 4t^2}\,dt \;\xrightarrow[\alpha\to\pi/2]{}\; \frac{(-1)^p \pi}{4^p} \cos\!\left(\tfrac{\pi}{8}\right). \tag{3} \]

The right-hand side alternates in sign with \(p\). This alternating sign will be the source of the contradiction.

3. Deriving a contradiction

Assume, for contradiction, that \(\Xi(t)\) has only finitely many real zeros. Then there exists \(T > 1\) such that \(\Xi(t)\) keeps a fixed sign for all \(t > T\); take this sign to be positive.

In that case, the integral on the left-hand side of (3) becomes strictly positive for all sufficiently large \(p\), because the factor \(t^{2p}\) forces the main contribution to come from large \(t\), where \(\Xi(2t) > 0\). We obtain upper and lower bounds of the form

\[ C_1 T^{2p} \;<\; \int_T^\infty \frac{(e^{\alpha t}+e^{-\alpha t})\,t^{2p}\Xi(2t)} {\tfrac14+4t^2}\,dt \;<\; C_2 T^{2p}, \]

with positive constants \(C_1, C_2\) independent of \(p\). Thus the integral is positive for all large \(p\).

But equation (3) says that the same integral tends to \[ \frac{(-1)^p\pi}{4^p}\cos\!\left(\tfrac{\pi}{8}\right), \] which changes sign as \(p\) increases. This contradiction proves the result.

Main result

Theorem (Hardy, 1914). The Riemann zeta function has infinitely many zeros on the line \(\sigma=\tfrac12\). Equivalently, the real entire function \(\Xi(t)\) has infinitely many real zeros.

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