Discrete universality of the Riemann zeta-function in short intervals
نویسندگان
چکیده
منابع مشابه
On the Divisor Function and the Riemann Zeta-function in Short Intervals
We obtain, for T ε ≤ U = U(T ) ≤ T 1/2−ε, asymptotic formulas for Z 2T T (E(t+ U)− E(t)) dt, Z 2T T (∆(t+ U)−∆(t)) dt, where ∆(x) is the error term in the classical divisor problem, and E(T ) is the error term in the mean square formula for |ζ( 1 2 + it)|. Upper bounds of the form Oε(T 1+εU2) for the above integrals with biquadrates instead of square are shown to hold for T 3/8 ≤ U = U(T ) ≪ T ...
متن کاملOn the Mean Values of the Riemann Zeta-function in Short Intervals
It is proved that, for T ε ≤ G = G(T) ≤
متن کاملOn the mean square of the Riemann zeta-function in short intervals
It is proved that, for T ε 6 G = G(T ) 6 2 √ T , ∫︁ 2T T (︁ I1(t+G,G)− I1(t, G) )︁2 dt = TG 3 ∑︁ j=0 aj log (︁√ T G )︁ +Oε(T 1+εG1/2 + T 1/2+εG2) with some explicitly computable constants aj (a3 > 0) where, for fixed k ∈ N, Ik(t, G) = 1 √ π ∫︁ ∞ −∞ |ζ( 1 2 + it+ iu)| 2ke−(u/G) 2 du. The generalizations to the mean square of I1(t+U,G)−I1(t, G) over [T, T+H] and the estimation of the mean square ...
متن کاملAn Effective Universality Theorem for the Riemann Zeta Function
Let 0 < r < 1/4, and f be a non-vanishing continuous function in |z| ≤ r, that is analytic in the interior. Voronin’s universality theorem asserts that translates of the Riemann zeta function ζ(3/4 + z + it) can approximate f uniformly in |z| < r to any given precision ε, and moreover that the set of such t ∈ [0, T ] has measure at least c(ε)T for some c(ε) > 0, once T is large enough. This was...
متن کاملPrimes in almost all short intervals and the distribution of the zeros of the Riemann zeta-function
Abstract We study the relations between the distribution of the zeros of the Riemann zeta-function and the distribution of primes in “almost all” short intervals. It is well known that a relation like ψ(x)−ψ(x−y) ∼ y holds for almost all x ∈ [N, 2N ] in a range for y that depends on the width of the available zero-free regions for the Riemann zeta-function, and also on the strength of density b...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Applicable Analysis and Discrete Mathematics
سال: 2020
ISSN: 1452-8630,2406-100X
DOI: 10.2298/aadm190704019l