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arXiv · 2403.00902

Sharp bounds for joint moments of the Riemann zeta function

Michael J. Curran·André Heycock
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Abstract

In previous work, the first author obtained conjecturally sharp upper bounds for the joint moments of the $(2k-2h)^{\text{th}}$ power of the Riemann zeta function with the $2h^{\text{th}}$ power of its derivative on the critical line in the range $1\leq k \leq 2$, $0 \leq h \leq 1$. Unconditionally, we extend these upper bounds to all $0 \leq h\leq k \leq 2$, and obtain lower bounds for all $0\leq h \leq k+1/2$. Assuming the Riemann hypothesis, we give sharp bounds for all $0\leq h \leq k$. We also prove upper bounds of the conjectured order for more general joint moments of zeta with its higher derivatives.

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Michael J. Curran, André Heycock. 2024-03-01. Sharp bounds for joint moments of the Riemann zeta function. https://arxiv.org/abs/2403.00902

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