Let \(H_0\) denote a verification height for RH. Let \(10^9/H_0≤ k \leq 1\), \(t {\gt} 0\), \(H \in [1002, H_0)\), \(α {\gt} 0\), \(δ ≥ 1\), \(\eta _0 = 0.23622\), \(1 + \eta _0 \leq \mu \leq 1+\eta \), and \(\eta \in (\eta _0, 1/2)\) be fixed. Let \(\sigma {\gt} 1/2 + d / \log H_0\). Then for any \(T \geq H_0\), one has

and

.

If \(|N(T) - (T/2\pi \log (T/2\pi e) + 7/8)| \leq R(T)\) then \(\sum _{U \leq \gamma {\lt} V} 1/\gamma \leq B_1(U,V)\).

For each pair \(T_0,S_0\) in Table 1 we have, for all \(V {\gt} T_0\), \(\sum _{0 {\lt} \gamma {\lt} V} 1/\gamma {\lt} S_0 + B_1(T_0,V)\).

Let \(T_0 \geq 2\) and \(\gamma {\gt} 0\). Assume that there exist \(c_1, c_2, p, q, T_0\) for which one has a zero density bound. Assume \(\sigma \geq 5/8\) and \(T_0 \leq U {\lt} V\). Then \(s_0(σ,U,V) \leq B_0(\sigma ,U,V)\).

\(\Gamma (3,x) = (x^2 + 2(x+1)) e^{-x}\).

For \(s{\gt}1\), one has \(\Gamma (s,x) \sim x^{s-1} e^{-x}\).

Let \(x {\gt} e^{50}\) and \(50 {\lt} T {\lt} x\). Then \(E_\psi (x) \leq \sum _{|\gamma | {\lt} T} |x^{\rho -1}/\rho | + 2 \log ^2 x / T\).

For any \(\alpha \in (0,1/2]\) and \(\omega \in [0,1]\) there exist \(M, x_M\) such that for \(\max (51, \log x) {\lt} T {\lt} (x^\alpha -2)/5\) and some \(T^* \in [T, 2.45 T]\),

for all \(x ≥ x_M\).

Let \(x {\gt} e^{50}\) and \(3 \log x {\lt} T {\lt} \sqrt{x}/3\). Then \(E_\psi (x) ≤ \sum _{|\gamma | {\lt} T} |x^{\rho -1}/\rho | + 2 \log ^2 x / T\).

Let \(\sigma _1 \in (1/2,1)\) and let \((T_0,S_0)\) be taken from Table 1. Then \(\Sigma _0^{\sigma _1} ≤ 2 x^{-1/2} (S_0 + B_1(T_0,T)) + (x_1^{\sigma _1-1} - x^{-1/2}) B_1(H_0,T)\).

\(\Sigma _a^b = 2 * \sum _{H_a ≤ \gamma {\lt} T; a \leq \beta {\lt} b} \frac{x^{\beta -1}}{\gamma }\).

If \(\sigma {\lt} 1 - 1/R \log H_0\) then \(H_σ = H_0\).

Let \(N \geq 2\) be an integer. If \(5/8 \leq \sigma _1 {\lt} \sigma _2 \leq 1\), \(T \geq H_0\), then \(\Sigma _{\sigma _1}^{\sigma _2} ≤ 2 x^{-(1-\sigma _1)+(\sigma _2-\sigma _1/N)}B_0(\sigma _1, H_{\sigma _1}, T) + 2 x^{(1-\sigma _1)} (1 - x^{-(\sigma _2-\sigma _1)/N}) \sum _{n=1}^{N-1} B_0(\sigma ^{(n)}, H^{(n)}, T) x^{(\sigma _2-\sigma _1) (n+1)/N}\).

If \(\sigma _1 \geq 0.9\) then \(\Sigma _{\sigma _1}^{\sigma _2} \leq 0.00125994 x^{\sigma _2-1}\).

Let \(5/8 {\lt} \sigma _2 \leq 1\), \(t_0 = t_0(\sigma _2,x) = \max (H_{\sigma _2}, \exp ( \sqrt{\log x}/R))\) and \(T {\gt} 0\). Let \(K \geq 2\) and consider a strictly increasing sequence \((t_k)_{k=0}^K\) such that \(t_k = T\). Then \(\Sigma _{\sigma _2}^1 ≤ 2 N(\sigma _2,T) x^{-1/R\log t_0}/t_0\) and \(\Sigma _{\sigma _2}^1 ≤ 2 ((\sum _{k=1}^{K-1} N(\sigma _2, t_k) (x^{-1/R\log t_{k-1}} / t_{k-1} - x^{-1/(R \log t_k)}/t_k)) + x^{-1/R \log t_{K-1}}/t_{K-1} N(\sigma _2,T))\).

Let \(5/8 {\lt} \sigma _2 \leq 1\), \(t_0 = t_0(\sigma _2, x) = \max \left(H_{\sigma _2}, \exp \left(\sqrt{\frac{\log x}{R}}\right)\right)\), \(T {\gt} t_0\). Let \(K \geq 2\), \(\lambda = (T/t_0)^{1/K}\), and consider \((t_k)_{k=0}^K\) the sequence given by \(t_k = t_0 \lambda ^k\). Then

where

and \(\tilde{N}(\sigma , T)\) satisfy (ZDB) \(N(\sigma , T) \leq \tilde{N}(\sigma , T)\).

Fix \(K \geq 2\) and \(c {\gt} 1\), and set \(t_0\), \(T\), and \(\sigma _2\) as functions of \(x\) defined by

Then, with \(\varepsilon _4(x, \sigma _2, K, T)\) as defined in (3.22), we have that as \(x \to \infty \),

where \(c_1\) is an admissible value for (ZDB) on some interval \([\sigma _1, 1]\). Moreover, both \(\varepsilon _4(x, \sigma _2, K, T)\) and \(\frac{\varepsilon _4(x, \sigma _2, K, T) t_0^2}{(\log t_0)^3}\) are decreasing in \(x\) for \(x {\gt} \exp (Re^2)\).

For any \(x_0\) with \(\log x_0 {\gt} 1000\), and all \(0.9 {\lt} \sigma _2 {\lt} 1\), \(2 \leq c \leq 30\), and \(N, K \geq 1\) the formula \(\varepsilon (x_0) := \varepsilon (x_0, \sigma _2, c, N, K)\) as defined in (4.1) gives an effectively computable bound

Moreover, a collection of values, \(\varepsilon (x_0)\) computed with well chosen parameters are provided in Table 5.

For all \(0 {\lt} \log x \leq 2100\) we have that

For all \(2100 {\lt} \log x \leq 200000\) we have that

If \(\log x_0 \geq 1000\) then we have an admissible bound for \(E_\psi \) with the indicated choice of \(A(x_0)\), \(B = 3/2\), \(C = 2\), and \(R = 5.5666305\).

For all x > 2 we have \(E_ψ(x) \leq 121.096 (\log x/R)^{3/2} \exp (-2 \sqrt{\log x/R})\) with \(R = 5.5666305\).

For all x > 2 we have \(E_ψ(x) \leq 9.22022(\log x)^{3/2} \exp (-0.8476836 \sqrt{\log x})\).

Let \(b_1, b_2\) satisfy \(1000 \leq b_1 {\lt} b_2\). Let \(0.001 \leq \delta \leq 0.025\), \(\lambda {\gt} 1\), \(H {\lt} T {\lt} e^{b_1}\), and \(K = \left\lfloor \frac{\log \frac{T}{H}}{\log \lambda } \right\rfloor + 1\). Then for all \(x \in [e^{b_1}, e^{b_2}]\)

where \(s_0, s_1, s_2\) are respectively defined in Definitions 8.5.1, 8.5.4, and 8.5.5

Let \(x_0 \geq 1000\) and let \(\sigma \in [0.75, 1)\). For all \(x \geq e^{x_0}\),

where \(A\), \(B\), and \(C\) are defined in Definitions 8.5.10, 8.5.11.

For \(100 \leq x \leq 10^{19}\), one has

Let \(0 {\lt} \varepsilon {\lt} 10^{-3}\), \(c \geq 3\), \(x_0 \geq 100\) and \(\alpha \in [0, 1)\) such that the inequality

holds. We denote the zeros of the Riemann zeta function by \(\rho = \beta + i\gamma \) with \(\beta , \gamma \in \mathbb {R}\). Then, if \(\beta = \frac{1}{2}\) holds for \(0 {\lt} \gamma \leq \frac{c}{\varepsilon }\), the inequality

holds for all \(x \geq e^{\varepsilon \alpha } x_0\), where

The \(\nu _c(\alpha ) = \nu _{c,1}(\alpha )\) and \(\mu _c(\alpha ) = \mu _{c,1}(\alpha )\) where \(\nu _{c,\varepsilon }(\alpha )\) and \(\mu _{c,\varepsilon }(\alpha )\) are defined by [ 3 , p. 2490 ] .

If \(b{\gt}0\) then \(|\psi (x) - x| \leq \varepsilon (b) x\) for all \(x \geq \exp (b)\), where \(\varepsilon \) is as in [ 2 , Table 8 ] .

\(F_{+,\lambda }(y) \geq I_\lambda (y)\) for all \(y\).

\(F_{+,\lambda }(y) \geq I_\lambda (y)\) for all \(y\).

\(\int (F_{+,\lambda }(y)-I_\lambda (y))\ dy = \frac{1}{1-e^{-|\lambda |}} - \frac{1}{|\lambda |}\).

\(\int (I_\lambda (y) - F_{-,\lambda }(y))\ dy = \frac{1}{|\lambda |} - \frac{1}{e^{|\lambda |} - 1}\).