Hardy–Littlewood zeta-function conjectures

In mathematics, the Hardy–Littlewood zeta-function conjectures, named after Godfrey Harold Hardy and John Edensor Littlewood, are two conjectures concerning the distances between zeros and the density of zeros of the Riemann zeta function.

## Conjectures[edit]

In 1914, Godfrey Harold Hardy proved[1] that the Riemann zeta function  ζ ( 1 2 + i t ) {\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}} has infinitely many real zeros.

Let  N ( T ) {\displaystyle N(T)} be the total number of real zeros,  N 0 ( T ) {\displaystyle N_{0}(T)} be the total number of zeros of odd order of the function  ζ ( 1 2 + i t ) {\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}} , lying on the interval  ( 0 , T ] {\displaystyle (0,T]} .

Hardy and Littlewood claimed[2] two conjectures. These conjectures – on the distance between real zeros of  ζ ( 1 2 + i t ) {\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}} and on the density of zeros of  ζ ( 1 2 + i t ) {\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}} on intervals  ( T , T + H ] {\displaystyle (T,T+H]} for sufficiently great  T > 0 {\displaystyle T>0} ,  H = T a + ε {\displaystyle H=T^{a+\varepsilon }} and with as less as possible value of  a > 0 {\displaystyle a>0} , where  ε > 0 {\displaystyle \varepsilon >0} is an arbitrarily small number – open two new directions in the investigation of the Riemann zeta function.

1\. For any  ε > 0 {\displaystyle \varepsilon >0} there exists such  T 0 = T 0 ( ε ) > 0 {\displaystyle T_{0}=T_{0}(\varepsilon )>0} that for  T ≥ T 0 {\displaystyle T\geq T_{0}} and  H = T 0.25 + ε {\displaystyle H=T^{0.25+\varepsilon }} the interval  ( T , T + H ] {\displaystyle (T,T+H]} contains a zero of odd order of the function  ζ ( 1 2 + i t ) {\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}} .

2\. For any  ε > 0 {\displaystyle \varepsilon >0} there exist  T 0 = T 0 ( ε ) > 0 {\displaystyle T_{0}=T_{0}(\varepsilon )>0} and  c = c ( ε ) > 0 {\displaystyle c=c(\varepsilon )>0} , such that for  T ≥ T 0 {\displaystyle T\geq T_{0}} and  H = T 0.5 + ε {\displaystyle H=T^{0.5+\varepsilon }} the inequality  N 0 ( T + H ) − N 0 ( T ) ≥ c H {\displaystyle N_{0}(T+H)-N_{0}(T)\geq cH} is true.

## Status[edit]

In 1942, Atle Selberg studied the problem 2 and proved that for any  ε > 0 {\displaystyle \varepsilon >0} there exists such  T 0 = T 0 ( ε ) > 0 {\displaystyle T_{0}=T_{0}(\varepsilon )>0} and  c = c ( ε ) > 0 {\displaystyle c=c(\varepsilon )>0} , such that for  T ≥ T 0 {\displaystyle T\geq T_{0}} and  H = T 0.5 + ε {\displaystyle H=T^{0.5+\varepsilon }} the inequality  N ( T + H ) − N ( T ) ≥ c H log ⁡ T {\displaystyle N(T+H)-N(T)\geq cH\log T} is true.

In his turn, Selberg made his conjecture[3] that it's possible to decrease the value of the exponent  a = 0.5 {\displaystyle a=0.5} for  H = T 0.5 + ε {\displaystyle H=T^{0.5+\varepsilon }} which was proved 42 years later by A.A. Karatsuba.[4]

## References[edit]

1. ^ Hardy, G.H. (1914). "Sur les zeros de la fonction  ζ ( s ) {\displaystyle \zeta (s)} ". Compt. Rend. Acad. Sci. 158: 1012–1014.
2. ^ Hardy, G.H.; Littlewood, J.E. (1921). "The zeros of Riemann's zeta-function on the critical line". Math. Z. 10 (3–4): 283–317. doi:10.1007/bf01211614. S2CID 126338046.
3. ^ Selberg, A. (1942). "On the zeros of Riemann's zeta-function". SHR. Norske Vid. Akad. Oslo. 10: 1–59.
4. ^ Karatsuba, A. A. (1984). "On the zeros of the function ζ(s) on short intervals of the critical line". Izv. Akad. Nauk SSSR, Ser. Mat. 48 (3): 569–584.