Moments of Hardy's function over short intervals

2016/12/06 by Aleksandar Ivić, Ivić, Aleksandar
Mathematics · #Advanced Harmonic Analysis Research #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1612.01698

openalex publication_date 2016/12/06 · openalex created_date 2017/05/19 · openalex updated_date 2026/07/28

Abstract

Let as usual Z(t) = ζ(1/2+it)χ^-1/2(1/2+it) denote Hardy's function, where ζ(s) = χ(s)ζ(1-s). Assuming the Riemann hypothesis upper and lower bounds for some integrals involving Z(t) and Z'(t) are proved. It is also proved that H(log T)^k2 ≪_k,α ∑_T1 is a fixed integer, γ, γ⁺ denote ordinates of consecutive complex zeros of ζ(s) and T^α≤ H ≤ T, where α is a fixed constant such that 0

Citations

• On some reasons for doubting the Riemann hypothesis

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