scholarly journals High-Dimensional D. H. Lehmer Problem over Quarter Intervals

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Tianping Zhang
Keyword(s):  
Error Term ◽  
High Dimensional ◽  
Character Sum ◽  
Mean Square ◽  
Lehmer Problem ◽  
The Mean

The high-dimensional D. H. Lehmer problem over quarter intervals is studied. By using the properties of character sum and the estimates of DirichletL-function, the previous result is improved to be the best possible in the case ofq = p, an odd prime withp≡1(mod 4), which is shown by studying the mean square value of the error term.

scholarly journals The asymptotic behaviour of the mean-square of fractional part sums

2000 ◽  
Vol 43 (2) ◽  
pp. 309-323 ◽  
Author(s):  
Manfred Kühleitner ◽  
Werner Georg Nowak
Keyword(s):  
Asymptotic Formula ◽  
Asymptotic Behaviour ◽  
Error Term ◽  
Fractional Part ◽  
Mean Square ◽  
The Mean

AbstractIn this article we consider sums S(t) = Σnψ (tf(n/t)), where ψ denotes, essentially, the fractional part minus ½ f is a C4-function with f″ non-vanishing, and summation is extended over an interval of order t. For the mean-square of S(t), an asymptotic formula is established. If f is algebraic this can be sharpened by the indication of an error term.

On Riesz means of the coefficients of the Rankin–Selberg series

1999 ◽  
Vol 127 (1) ◽  
pp. 117-131 ◽  
Author(s):  
ALEKSANDAR IVIĆ ◽  
KOHJI MATSUMOTO ◽  
YOSHIO TANIGAWA
Keyword(s):  
Asymptotic Formula ◽  
Upper Bound ◽  
Error Term ◽  
Weighted Sum ◽  
Mean Square ◽  
Riesz Means ◽  
The Mean ◽  
Voronoi Formula

We study Δ(x; ϕ), the error term in the asymptotic formula for [sum ]n[les ]xcn, where the cns are generated by the Rankin–Selberg series. Our main tools are Voronoï-type formulae. First we reduce the evaluation of Δ(x; ϕ) to that of Δ1(x; ϕ), the error term of the weighted sum [sum ]n[les ]x(x−n)cn. Then we prove an upper bound and a sharp mean square formula for Δ1(x; ϕ), by applying the Voronoï formula of Meurman's type. We also prove that an improvement of the error term in the mean square formula would imply an improvement of the upper bound of Δ(x; ϕ). Some other related topics are also discussed.

On the mean square of an arithmetical error term of the Selberg class in short intervals

2016 ◽  
Vol 12 (06) ◽  
pp. 1675-1701
Author(s):  
Xiaodong Cao ◽  
Yoshio Tanigawa ◽  
Wenguang Zhai
Keyword(s):  
Dirichlet Series ◽  
Error Term ◽  
Selberg Class ◽  
Mean Square ◽  
Short Intervals ◽  
Type Formula ◽  
The Mean

Let [Formula: see text] be a Dirichlet series in the Selberg class of degree [Formula: see text] and let [Formula: see text] be the arithmetical error term of [Formula: see text]. We derive two kinds of the mean square estimates of [Formula: see text] in short intervals of Jutila’s type. Our method is based on the truncated Tong-type formula of [Formula: see text]. We also give several applications of these estimates in the arithmetical problems.

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