Return to Answer

For every $\varepsilon>0$, there exist $c(\varepsilon)>0$ and $X_0(\varepsilon)>0$ such that for any $X>X_0(\varepsilon)$ we have $$\sup_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}>c(\varepsilon)\qquad\text{and}\qquad \inf_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}<-c(\varepsilon).$$ More precisely, Ingham (1935) proved a stronger result for the case when the real parts of the zeta zeros have a maximum, while Pintz (1980) proved a stronger result for the case when the real parts of the zeta zeros do not have a maximum. There might be even stronger results in the literature, please check.

For any $\varepsilon>0$, there exist $c(\varepsilon)>0$ and $X_0(\varepsilon)>0$ such that for any $X>X_0(\varepsilon)$ we have $$\sup_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}>c(\varepsilon)\qquad\text{and}\qquad \inf_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}<-c(\varepsilon).$$ More precisely, Ingham (1935) proved a stronger result for the case when the real parts of the zeta zeros have a maximum, while Pintz (1980) proved a stronger result for the case when the real parts of the zeta zeros do not have a maximum. There might be even stronger results in the literature, please check.

For every $\varepsilon>0$, there exists $X_0(\varepsilon)>0$ and $c(\varepsilon)>0$ such that for any $X>X_0(\varepsilon)$ we have $$\sup_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}>c(\varepsilon)\qquad\text{and}\qquad \inf_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}<-c(\varepsilon).$$ More precisely, Ingham (1935) proved a stronger result for the case when the real parts of the zeta zeros have a maximum, while Pintz (1980) proved a stronger result for the case when the real parts of the zeta zeros do not have a maximum. There might be even stronger results in the literature, please check.

For every $\varepsilon>0$, there exist $c(\varepsilon)>0$ and $X_0(\varepsilon)>0$ such that for any $X>X_0(\varepsilon)$ we have $$\sup_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}>c(\varepsilon)\qquad\text{and}\qquad \inf_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}<-c(\varepsilon).$$ More precisely, Ingham (1935) proved a stronger result for the case when the real parts of the zeta zeros have a maximum, while Pintz (1980) proved a stronger result for the case when the real parts of the zeta zeros do not have a maximum. There might be even stronger results in the literature, please check.

For every $\varepsilon>0$, there exists $X_0(\varepsilon)>0$ such that for any $X>X_0(\varepsilon)$ we have $$\sup_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}>0>\inf_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}.$$ More precisely, Ingham (1935) proved a stronger result for the case when the real parts of the zeta zeros have a maximum, while Pintz (1980) proved a stronger result for the case when the real parts of the zeta zeros do not have a maximum. There might be even stronger results in the literature, please check.

For every $\varepsilon>0$, there exists $X_0(\varepsilon)>0$ and $c(\varepsilon)>0$ such that for any $X>X_0(\varepsilon)$ we have $$\sup_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}>c(\varepsilon)\qquad\text{and}\qquad \inf_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}<-c(\varepsilon).$$ More precisely, Ingham (1935) proved a stronger result for the case when the real parts of the zeta zeros have a maximum, while Pintz (1980) proved a stronger result for the case when the real parts of the zeta zeros do not have a maximum. There might be even stronger results in the literature, please check.