Selberg's zeta function conjecture

Last updated February 21, 2019

In mathematics, the Selberg conjecture, named after Atle Selberg, is a theorem about the density of zeros of the Riemann zeta function ζ(1/2 + it). It is known that the function has infinitely many zeroes on this line in the complex plane: the point at issue is how densely they are clustered. Results on this can be formulated in terms of N(T), the function counting zeroes on the line for which the value of t satisfies 0 ≤t≤T.

Atle Selberg was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory. He was awarded the Fields Medal in 1950.

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.

The Riemann zeta function or Euler–Riemann zeta function, $ζ (s)$ , is a function of a complex variable s that analytically continues the sum of the Dirichlet series

Background

In 1942 Atle Selberg investigated the problem of the Hardy–Littlewood conjecture 2; and he proved that for any

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.

\varepsilon >0

there exist

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

holds true.

In his turn, Selberg stated a conjecture relating to shorter intervals,^[1] namely that it is possible to decrease the value of the exponent a = 0.5 in

H=T^{0.5+\varepsilon }.

Proof of the conjecture

In 1984 Anatolii 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 in the ordinate t (T, T + H) contains at least cH ln T real zeros of the Riemann zeta function

\zeta {\Bigl (}{\tfrac {1}{2}}+it{\Bigr )};

and thereby confirmed the Selberg conjecture. The estimates of Selberg and Karatsuba cannot be improved in respect of the order of growth as T → +∞.

Further work

In 1992 Karatsuba proved^[5] that an analog of the Selberg conjecture holds for "almost all" intervals (T, T + H], H = T^ε, where ε is an arbitrarily small fixed positive number. The Karatsuba method permits one to investigate zeroes 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 ε, ε₁ satisfying the conditions 0 < ε, ε₁< 1 almost all intervals (T, T + H] for H ≥ exp[(ln T)^ε] contain at least H (ln T)^1 −ε₁ zeros of the function ζ(1/2 + it). This estimate is quite close to the conditional result that follows from the Riemann hypothesis.

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In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. It was proposed by Bernhard Riemann (1859), after whom it is named. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.

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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.

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[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.