Arithmetic equivalent of essential simplicity of zeta zeros

Abstract

Let R ( x ) R(x) and S ( t ) S(t) be the remainder terms in the prime number theorem and the Riemann-von Mangoldt formula respectively, that is ψ ( x ) = x + R ( x ) \psi (x) = x + R(x) and N ( t ) = ( 1 / 2 π ) ∫ 0 t log ⁡ ( τ / 2 π ) d τ + S ( t ) + 7 / 8 + O ( 1 / t ) N(t) = (1/2\pi )\int _0^t {\log (\tau /2\pi )\,d\tau + S(t) + 7/8 + O(1/t)} . We are interested in the following integrals: J ( T , β ) = ∫ 1 T β ( R ( x + x / T ) − R ( x ) ) 2 d x / x 2 J(T,\beta ) = \int _1^{{T^\beta }} {{{(R(x + x/T) - R(x))}^2}dx/{x^2}} and I ( T , α ) = ∫ 1 T ( S ( t + α / L ) − S ( t ) ) 2 d t I(T,\alpha ) = \int _1^T {{{(S(t + \alpha /L) - S(t))}^2}dt} , where L = ( 2 π ) − 1 log T L = {(2\pi )^{ - 1}}\log \,T . Furthermore, denote by N ( T , α ) ( N ∗ ( T ) ) N(T,\alpha )(N^{\ast }(T)) the number of pairs of zeros 1 2 + i Υ , 1 2 + i Υ ′ \frac {1} {2} + i\Upsilon ,\frac {1} {2} + i\Upsilon ’ with 0 > Υ ⩽ T 0 > \Upsilon \leqslant T and 0 > ( Υ ′ − Υ ) L ⩽ α ( ( Υ ′ − Υ ) L = 0 ) 0 > (\Upsilon ’ - \Upsilon )L \leqslant \alpha \,((\Upsilon ’ - \Upsilon )L = 0) —i.e., off-diagonal and diagonal pairs. Theorem. Assume the Riemann hypothesis. The following three hypotheses (A), (B) and ( C 1 , C 2 ) ({{\text {C}}_1},{{\text {C}}_2}) are equivalent: for β → ∞ \beta \to \infty and α → 0 \alpha \to 0 as T → ∞ T \to \infty we have (A) J ( T , β ) ∼ β T − 1 log 2 T J(T,\beta ) \sim \beta {T^{ - 1}}{\log ^2}T , (B) I ( T , α ) ∼ α T I(T,\alpha ) \sim \alpha T and ( C 1 ) N ∗ ( T ) ∼ T L , ( C 2 ) N ( T , α ) = o ( T L ) ({{\text {C}}_1})\;N^{\ast }(T) \sim TL,({{\text {C}}_2})N(T,\alpha ) = o(TL) . Hypothesis ( C 1 , C 2 ) ({{\text {C}}_1},{{\text {C}}_2}) is called the essential simplicity hypothesis.