Mean values of Hardy's function on the Gram lattices of arbitrary phase

Abstract

Let θ be the Riemann-Siegel theta function and Z(t) = e iθ(t) ζ(1 2 + it) Hardy's function. For α ∈ [0, 1) let L α = {t > t 0 : θ(t) &equiv; απ (mod π)}, numbered by θ(t n) = (n + α)π; L 0 is the sequence of Gram points. We prove that for every α T <tn≤2T ζ(1 2 + it n) 2θ ′ (t n) = 1 + e −2πiα T 2π + O T 1/4 log T , so that the θ ′-weighted mean of (−1) n Z(t n) over L α is 2 cos πα. At α = 0 this is Titch-marsh's theorem on the mean of Hardy's function at the Gram points, for the alternating sum (−1) n Z(g n) = ζ(1 2 + ig n), with an error that, combined with Bourgain's sub-convexity bound, is O(N 13/84+ε); the parity-separated sums over even and over odd Gram points, for which Cao, Tanigawa and Zhai have O(N 1/4 log 3/4 N log log N), are a finer statistic that the present method does not reach. At α = 1 2 , where ζ(1 2 + it n) is purely imaginary, the mean is 0. The proof is a one-page contour integration of ζ(s)/(1 + λχ(s)), λ = −e 2πiα , whose poles are exactly the points of L α on the critical line; the vertical edges of the contour lie in the half-planes of absolute convergence and contribute T and T /λ with error O(1). The theorem extends to a class of L-functions under explicit analytic hypotheses on the functional equation. We also report how Gram's law depends on the phase: on the first 600 zeros the proportion of lattice intervals containing exactly one zero falls from 0.936 at α = 0 to 0.484 at α = 1 2 , in the direction the mean value suggests.

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Cite

BibTeX RIS

BibTeX

@misc{pablo2026mean,
  title = {{Mean values of Hardy's function on the Gram lattices of arbitrary phase}},
  author = {Veltri Pablo},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.13140/rg.2.2.24583.36005},
  url = {https://zenodo.org/records/23769727},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Mean values of Hardy's function on the Gram lattices of arbitrary phase
AU  - Veltri Pablo
PY  - 2026
DA  - 2026/10/29
PB  - Zenodo
DO  - 10.13140/rg.2.2.24583.36005
UR  - https://zenodo.org/records/23769727
AB  - Let &theta; be the Riemann-Siegel theta function and Z(t) = e i&theta;(t) &zeta;(1 2 + it) Hardy's function. For α &isin; [0, 1) let L α = {t > t 0 : &theta;(t) &equiv; απ (mod π)}, numbered by &theta;(t n) = (n + α)π; L 0 is the sequence of Gram points. We prove that for every α T <tn≤2T &zeta;(1 2 + it n) 2&theta; ′ (t n) = 1 + e −2πiα T 2π + O T 1/4 log T , so that the &theta; ′-weighted mean of (−1) n Z(t n) over L α is 2 cos πα. At α = 0 this is Titch-marsh's theorem on the mean of Hardy's function at the Gram points, for the alternating sum (−1) n Z(g n) = &zeta;(1 2 + ig n), with an error that, combined with Bourgain's sub-convexity bound, is O(N 13/84+&epsilon;); the parity-separated sums over even and over odd Gram points, for which Cao, Tanigawa and Zhai have O(N 1/4 log 3/4 N log log N), are a finer statistic that the present method does not reach. At α = 1 2 , where &zeta;(1 2 + it n) is purely imaginary, the mean is 0. The proof is a one-page contour integration of &zeta;(s)/(1 + λ&chi;(s)), λ = −e 2πiα , whose poles are exactly the points of L α on the critical line; the vertical edges of the contour lie in the half-planes of absolute convergence and contribute T and T /λ with error O(1). The theorem extends to a class of L-functions under explicit analytic hypotheses on the functional equation. We also report how Gram's law depends on the phase: on the first 600 zeros the proportion of lattice intervals containing exactly one zero falls from 0.936 at α = 0 to 0.484 at α = 1 2 , in the direction the mean value suggests.
ER  -

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