On the error term of the fourth moment of the Riemann zeta-function

2025/06/20 by Palojärvi, Neea, Trudgian, Tim · 1 citation
#11M06 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.16766

Abstract

We examine the size of E₂(T), the error term in the asymptotic formula for ∫₀^T |ζ(1/2 + it)|⁴ dt where ζ(s) is the Riemann zeta-function. We make improvements in the powers of log T in the known bounds for E₂(T) and ∫₀^T E₂(t)² dt. As a consequence, we obtain small logarithmic improvements for kth moments where 8≤ k≤ 12. In particular, we make a modest improvement on the 12th power moment for ζ(s).

Citations

• Dirichlet L-functions on the critical line and multiplicative chaos
• Decoupling, exponential sums and the Riemann zeta function
• A note on the mean value of the zeta and L-functions. XIV
• Heegner points, cycles and Maass forms

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