Selberg's zeta function conjecture

In mathematics, the Selberg conjecture, named after Atle Selberg, is about the density of zeros of the Riemann zeta function $\zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}$ .

In 1942 Atle Selberg investigated the problem of the Hardy–Littlewood conjecture 2 and proved that for any $\varepsilon >0$ there exists such $T_{0}=T_{0}(\varepsilon )>0$ and $c=c(\varepsilon )>0$ , such that for $T\geq T_{0}$ and $H=T^{0.5+\varepsilon }$ the inequality $N(T+H)-N(T)\geq cH\log T$ is true.

In his turn, Selberg claim a conjecture^[1] that it's possible to decrease the value of the exponent $a=0.5$ for $H=T^{0.5+\varepsilon }$ .

In 1984 Anatolii Alexeevitch Karatsuba proved^[2]^[3]^[4] that for a fixed $\varepsilon$ satisfying the condition $0<\varepsilon <0.001$ , a sufficiently large $T$ and $H=T^{a+\varepsilon }$ , $a={\tfrac {27}{82}}={\tfrac {1}{3}}-{\tfrac {1}{246}}$ , the interval $(T,T+H)$ contains at least $cH\ln T$ real zeros of the Riemann zeta function $\zeta {\Bigl (}{\tfrac {1}{2}}+it{\Bigr )}$ and and therefore confirmed the Selberg conjecture.

The estimates of Atle Selberg and Karatsuba can not be improved in respect of the order of growth as $T\to +\infty$ .

In 1992 A.A. Karatsuba proved^[5] that an analog of the Selberg conjecture holds for «almost all» intervals $(T,T+H]$ , $H=T^{\varepsilon }$ , where $\varepsilon$ is an arbitrarily small fixed positive number. The Karatsuba method permits to investigate zeros of the Riemann zeta-function on «supershort» intervals of the critical line, that is, on the intervals $(T,T+H]$ , the length $H$ of which grows slower than any, even arbitrarily small degree $T$ . In particular, he proved that for any given numbers $\varepsilon$ , $\varepsilon _{1}$ satisfying the conditions $0<\varepsilon ,\varepsilon _{1}<1$ almost all intervals $(T,T+H]$ for $H\geq \exp {\{(\ln T)^{\varepsilon }\}}$ contain at least $H(\ln T)^{1-\varepsilon _{1}}$ zeros of the function $\zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}$ . This estimate is quite close to the one that follows from the Riemann hypothesis.

References

^ Selberg, A. (1942). "On the zeros of Riemann's zeta-function". Shr. Norske Vid. Akad. Oslo (10): 1–59.
^ 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.
^ Karatsuba, A. A. (1984). "The distribution of zeros of the function ζ(1/2 + it)". Izv. Akad. Nauk SSSR, Ser. Mat. (48:6): 1214–1224.
^ Karatsuba, A. A. (1985). "On the zeros of the Riemann zeta-function on the critical line". Proc. Steklov Inst. Math. (167): 167–178.
^ Karatsuba, A. A. (1992). "On the number of zeros of the Riemann zeta-function lying in almost all short intervals of the critical line". Izv. Ross. Akad. Nauk, Ser. Mat. (56:2): 372–397.

[1] Selberg, A. (1942). "On the zeros of Riemann's zeta-function". Shr. Norske Vid. Akad. Oslo (10): 1–59.

[2] 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.

[3] Karatsuba, A. A. (1984). "The distribution of zeros of the function ζ(1/2 + it)". Izv. Akad. Nauk SSSR, Ser. Mat. (48:6): 1214–1224.

[4] Karatsuba, A. A. (1985). "On the zeros of the Riemann zeta-function on the critical line". Proc. Steklov Inst. Math. (167): 167–178.

[5] Karatsuba, A. A. (1992). "On the number of zeros of the Riemann zeta-function lying in almost all short intervals of the critical line". Izv. Ross. Akad. Nauk, Ser. Mat. (56:2): 372–397.

[1]

[2]

[3]

[4]

[5]