scholarly journals On the divisor function and the Riemann zeta-function in short intervals

2008 ◽  
Vol 19 (2) ◽  
pp. 207-224 ◽  
Author(s):  
Aleksandar Ivić
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Divisor Function ◽  
Short Intervals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals The generalized divisor problem and the Riemann hypothesis

1991 ◽  
Vol 122 ◽  
pp. 149-159 ◽  
Author(s):  
Hideki Nakaya
Keyword(s):  
Zeta Function ◽  
Natural Number ◽  
Complex Number ◽  
Riemann Zeta Function ◽  
Multiplicative Function ◽  
Riemann Hypothesis ◽  
Divisor Problem ◽  
Divisor Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Let dz(n) be a multiplicative function defined bywhere s = σ + it, z is a. complex number, and ζ(s) is the Riemann zeta function. Here ζz(s) = exp(z log ζ(s)) and let log ζ(s) take real values for real s > 1. We note that if z is a natural number dz(n) coincides with the divisor function appearing in the Dirichlet-Piltz divisor problem, and d-1(n) with the Möbious function.

scholarly journals Moments of the Riemann zeta function on short intervals of the critical line

2021 ◽  
Vol 49 (6) ◽  
Author(s):  
Louis-Pierre Arguin ◽  
Frédéric Ouimet ◽  
Maksym Radziwiłł
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Short Intervals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals Discrete universality of the Riemann zeta-function in short intervals

2020 ◽  
pp. 19-19
Author(s):  
Antanas Laurincikas
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Short Intervals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Discrete Universality ◽  
Positive Integers

We consider the approximation of analytic functions by shifts of the Riemann zeta-function ?(s+ikh) with fixed h > 0 when positive integers k run over the interval [N,N+M], where N1/3(logN)26=15 ? M ? N, and prove that those k have a positive lower density as N ? ?. The same is true for some compositions. Two types of h > 0 are discussed separately.

scholarly journals On the mean square of the Riemann zeta-function in short intervals

2009 ◽  
Vol 85 (99) ◽  
pp. 1-17 ◽  
Author(s):  
Aleksandar Ivic
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Mean Square ◽  
Short Intervals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
The Mean

It is proved that, for T? ? G = G(T)? ??T, ?T2T(I1(t+G,G)- I1(t,G))2 dt = TG ?aj logj (?T/G)+ O?(T1+? G1/2+ +T1/2?G? with some explicitly computable constants aj(a3>0)where, for fixed K ? N, Ik(t,G)= 1/?? ? ? -? |?(1/2 + it + iu)|2k e -(u/G)?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 of I2(t+ U,G) - I2(t,G) are also discussed.

scholarly journals On the Selberg integral of the k-divisor function and the 2k-th moment of the Riemann zeta-function

2010 ◽  
Vol 88 (102) ◽  
pp. 99-110 ◽  
Author(s):  
Giovanni Coppola
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Divisor Function ◽  
Elementary Approach ◽  
Selberg Integral ◽  
Reverse Link ◽  
Recent Approach ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Simple Average

In the literature one can find links between the 2k-th moment of the Riemann zeta-function and averages involving dk(n), the divisor function generated by ?k(s). There are, in fact, two bounds: one for the 2k-th moment of ?(s) coming from a simple average of correlations of the dk; and the other, which is a more recent approach, for the Selberg integral involving dk(n), applying known bounds for the 2k-th moment of the zeta-function. Building on the former work, we apply an elementary approach (based on arithmetic averages) in order to get the reverse link to the second work; i.e., we obtain (conditional) bounds for the 2k-th moment of the zeta-function from the Selberg integral bounds involving dk(n).

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