Moments and subconvexity of the zeta function

TheoremTwelfth moment of the Riemann zeta function on the critical line#●ready to prove

For every \(\varepsilon > 0\) there is a constant \(C\) such that for every real \(T \ge 1\),

\[\int_1^T |\zeta(1/2 + it)|^{12} \, dt \le C T^{2+\varepsilon}.\]

This is Heath-Brown's bound in the form quoted by the sources.

public theorem_wanted riemannZeta_twelfth_moment_le {ε : ℝ} (hε : 0 < ε) :
    ∃ C : ℝ, ∀ T : ℝ, 1 ≤ T →
      ∫ t in (1 : ℝ)..T, ‖riemannZeta (1 / 2 + t * I)‖ ^ 12 ≤ C * T ^ (2 + ε)

Sources

Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Lemma 2.4, equation (9); Guodong Hua, On Higher Moments of Dirichlet Coefficients Attached to Symmetric Power L-Functions Over Certain Sequences of Positive Integers, INTEGERS 24 (2024), Article A28, Lemma 5, equation (21).

TheoremHeath-Brown's twelfth moment bound with the logarithmic factor#●ready to prove

There are constants \(C\) and \(T_0\) such that for every real \(T \ge T_0\),

\[\int_1^T |\zeta(1/2 + it)|^{12} \, dt \le C T^2 (\log T)^{17}.\]

public theorem_wanted riemannZeta_twelfth_moment_le_log :
    ∃ C T₀ : ℝ, ∀ T : ℝ, T₀ ≤ T →
      ∫ t in (1 : ℝ)..T, ‖riemannZeta (1 / 2 + t * I)‖ ^ 12 ≤ C * T ^ 2 * Real.log T ^ 17

Sources

M. S. Datt, Ekta Soni, and A. Sankaranarayanan, On the Asymptotic Formula for the Gcd-Sum Function, INTEGERS 26 (2026), Article A70, Lemma 4.

TheoremTwelfth moment of the Riemann zeta function on the line of real part 5/7#●ready to prove

For every \(\varepsilon > 0\) there is a constant \(C\) such that for every real \(T \ge 1\),

\[\int_0^T |\zeta(5/7 + it)|^{12} \, dt \le C T^{1+\varepsilon}.\]

public theorem_wanted riemannZeta_twelfth_moment_five_sevenths_le {ε : ℝ} (hε : 0 < ε) :
    ∃ C : ℝ, ∀ T : ℝ, 1 ≤ T →
      ∫ t in (0 : ℝ)..T, ‖riemannZeta (5 / 7 + t * I)‖ ^ 12 ≤ C * T ^ (1 + ε)

Sources

M. S. Datt, Ekta Soni, and A. Sankaranarayanan, On the Asymptotic Formula for the Gcd-Sum Function, INTEGERS 26 (2026), Article A70, Lemma 1.

TheoremSubconvexity bound for the Riemann zeta function#●ready to prove

For every \(\varepsilon > 0\) there is a constant \(C\) such that for all real \(\sigma\) and \(t\) with \(1/2 \le \sigma \le 1 + \varepsilon\) and \(|t| \ge 1\),

\[|\zeta(\sigma + it)| \le C (1 + |t|)^{\max(13(1 - \sigma)/42,\, 0) + \varepsilon}.\]

The exponent comes from the exponent \(13/84\) on the critical line; the truncation at \(0\) follows both cited equations.

public theorem_wanted riemannZeta_norm_le_subconvex {ε : ℝ} (hε : 0 < ε) :
    ∃ C : ℝ, ∀ σ t : ℝ, 1 / 2 ≤ σ → σ ≤ 1 + ε → 1 ≤ |t| →
      ‖riemannZeta (σ + t * I)‖ ≤ C * (1 + |t|) ^ (max (13 / 42 * (1 - σ)) 0 + ε)

Sources

Guodong Hua, The General Divisor Problem of Cusp Form Coefficients over Arithmetic Progressions, INTEGERS 24 (2024), Article A3, Lemma 2.4, equation (10); Guodong Hua, On Higher Moments of Dirichlet Coefficients Attached to Symmetric Power L-Functions Over Certain Sequences of Positive Integers, INTEGERS 24 (2024), Article A28, Lemma 5, equation (22).

TheoremSubconvexity bound for the Dirichlet L-function modulo 4#●ready to prove

Let \(\varepsilon > 0\) and let \(\chi\) be a non-principal Dirichlet character modulo \(4\) (there is exactly one). Then there is a constant \(C\) such that for all real \(\sigma\) and \(t\) with \(1/2 \le \sigma \le 1 + \varepsilon\) and \(|t| \ge 1\),

\[|L(\sigma + it, \chi)| \le C (1 + |t|)^{\max((1 - \sigma)/3,\, 0) + \varepsilon}.\]

The truncation at \(0\) follows equation (23) of the source.

public theorem_wanted dirichletLFunction_mod_four_norm_le_subconvex {ε : ℝ} (hε : 0 < ε)
    (χ : DirichletCharacter ℂ 4) (hχ : χ ≠ 1) :
    ∃ C : ℝ, ∀ σ t : ℝ, 1 / 2 ≤ σ → σ ≤ 1 + ε → 1 ≤ |t| →
      ‖χ.LFunction (σ + t * I)‖ ≤ C * (1 + |t|) ^ (max (1 / 3 * (1 - σ)) 0 + ε)

Sources

Guodong Hua, On Higher Moments of Dirichlet Coefficients Attached to Symmetric Power L-Functions Over Certain Sequences of Positive Integers, INTEGERS 24 (2024), Article A28, Lemma 5, equation (23).

5 ready to prove

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