# On the mod-Gaussian convergence of a sum over primes

Wahl, Martin (2014). On the mod-Gaussian convergence of a sum over primes. Mathematische Zeitschrift, 276(3-4):635-654.

## Abstract

We prove mod-Gaussian convergence for a Dirichlet polynomial which approximates $${\mathrm{Im }}\log \zeta (1/2+it)$$ Im log ζ ( 1 / 2 + i t ) . This Dirichlet polynomial is sufficiently long to deduce Selberg's central limit theorem with an explicit error term. Moreover, assuming the Riemann hypothesis, we apply the theory of the Riemann zeta-function to extend this mod-Gaussian convergence to the complex plane. From this we obtain that $${\mathrm{Im }}\log \zeta (1/2+it)$$ Im log ζ ( 1 / 2 + i t ) satisfies a large deviation principle on the critical line. Results about the moments of the Riemann zeta-function follow.

## Abstract

We prove mod-Gaussian convergence for a Dirichlet polynomial which approximates $${\mathrm{Im }}\log \zeta (1/2+it)$$ Im log ζ ( 1 / 2 + i t ) . This Dirichlet polynomial is sufficiently long to deduce Selberg's central limit theorem with an explicit error term. Moreover, assuming the Riemann hypothesis, we apply the theory of the Riemann zeta-function to extend this mod-Gaussian convergence to the complex plane. From this we obtain that $${\mathrm{Im }}\log \zeta (1/2+it)$$ Im log ζ ( 1 / 2 + i t ) satisfies a large deviation principle on the critical line. Results about the moments of the Riemann zeta-function follow.

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