A lower bound for the sixth moment of the Riemann zeta function

Niklas Fraccaro

A lower bound for the sixth moment of the Riemann zeta function

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Niklas Fraccaro ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We prove that the lower limit of the sixth moment of the Riemann zeta function, normalized by

(a_3/9!)T(\log T)^9

, is greater than

41.007

. Our approach extends the two-piece amplifier construction used by Durkan and Page by allowing independent profiles, centered derivatives, and finite real linear combinations of two projected test families. An asymmetric twisted fourth-moment estimate and their amplified moment formula yield a common numerator kernel. Explicit balanced and unbalanced kernels determine the projected Gram form. A rational certificate with 250 tests, followed by approximation by admissible profiles, gives the stated bound.

Provenance statement

The original proof strategy and initial mathematical argument were developed by the author. OpenAI's GPT-6 was subsequently used to identify and correct a significant error in Theorem 2.1, assist with mathematical computations, derive alternative and more concise formulas, and simplify and improve parts of the proof and its presentation.

Tools used

OpenAI: ChatGPTVersion Astra-6

References

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Version history

v1Submitted by Niklas FraccaroInitial depositCurrentOct 28, 2026