scholarly journals On the mean square of the Riemann zeta function and the divisor problem

2008 ◽  
Vol 83 (97) ◽  
pp. 71-86
Author(s):  
Yifan Yang
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Integral Transforms ◽  
Divisor Problem ◽  
Mean Square ◽  
The Riemann Zeta Function ◽  
Dirichlet Divisor Problem ◽  
Error Terms ◽  
Riemann Zeta ◽  
The Mean

Let ?(T) and E(T) be the error terms in the classical Dirichlet divisor problem and in the asymptotic formula for the mean square of the Riemann zeta function in the critical strip, respectively. We show that ?(T) and E(T) are asymptotic integral transforms of each other. We then use this integral representation of ?(T) to give a new proof of a result of M. Jutila.

scholarly journals On the mean square formula of the error term in the Dirichlet divisor problem

2009 ◽  
Vol 146 (2) ◽  
pp. 277-287 ◽  
Author(s):  
YUK-KAM LAU ◽  
KAI-MAN TSANG
Keyword(s):  
Remainder Term ◽  
Error Term ◽  
Divisor Problem ◽  
Mean Square ◽  
The Riemann Zeta Function ◽  
Dirichlet Divisor Problem ◽  
Error Terms ◽  
Riemann Zeta ◽  
The Mean ◽  
Upper Estimate

AbstractLet F(x) be the remainder term in the mean square formula of the error term Δ(t) in the Dirichlet divisor problem. We improve on the upper estimate of F(x) obtained by Preissmann around twenty years ago. The method is robust, which applies to the same problem for the error terms in the circle problem and the mean square formula of the Riemann zeta-function.

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.

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