The Riemann and Lindelöf hypotheses are determined by thin sets of primes

Abstract:

Gonek, Graham and Lee have recently formulated an extension of the classical Lindelöf hypothesis. For various sets $\mathcal {N}$, their hypothesis $\operatorname {LH}[\,\mathcal {N}\,]$ asserts the existence of a smooth approximation $M(x)$ to the counting function of $\mathcal {N}$ that satisfies a specific set of properties. Let $\operatorname {LH}[\,\mathcal {N},M(x)\,]$ be the statement that $\operatorname {LH}[\,\mathcal {N}\,]$ holds with the function $M(x)$. For $\mathcal {N}≔\mathbb {P}$, the set of prime numbers, they have shown that the statement $\operatorname {LH}[\,\mathbb {P},\mathrm {li}(x)\,]$ is equivalent to the Riemann hypothesis (RH).

In this note, we find sets of primes $\mathcal {P}$, of small relative density $\varepsilon$ in $\mathbb {P}$, such that $\operatorname {LH}[\,\mathcal {P},\varepsilon \,\mathrm {li}(x)\,]$ is equivalent to RH. We also find thinner sets $\mathcal {P}\subset \mathbb {P}$, the thinnest ones satisfying \begin{equation*} \big |\{p\leqslant x:p\in \mathcal {P}\}\big |\sim M(x)\leqslant x^{1/2+o(1)} \qquad (x\to \infty ), \end{equation*} such that $\operatorname {LH}[\,\mathcal {P},M(x)\,]$ is equivalent to the nonvanishing of the Riemann zeta function in some half-plane $\{\sigma >\sigma _0\}$ with $\sigma _0<1$. For the latter sets $\mathcal {P}$, it is shown that a stronger variant $\operatorname {LH}^\star [\,\mathcal {P},M(x)\,]$ of the Lindelöf hypothesis holds if and only if RH is true.