scholarly journals On a Cochrane sum and its hybrid mean value formula (II)

2002 ◽  
Vol 276 (1) ◽  
pp. 446-457 ◽  
Author(s):  
Wenpeng Zhang
Keyword(s):  
Mean Value ◽  
Hybrid Mean Value ◽  
Mean Value Formula ◽  
Cochrane Sum

On a Generalized Cochrane Sum and Its Hybrid Mean Value Formula

2005 ◽  
Vol 9 (3) ◽  
pp. 373-380 ◽  
Author(s):  
Liu Hongyan ◽  
Zhang Wenpeng
Keyword(s):  
Mean Value ◽  
Hybrid Mean Value ◽  
Mean Value Formula ◽  
Cochrane Sum

scholarly journals Almansi Theorem and Mean Value Formula for Quaternionic Slice-regular Functions

2020 ◽  
Vol 30 (4) ◽  
Author(s):  
Alessandro Perotti
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Zonal Harmonic ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Quaternionic Polynomials ◽  
Mean Value Formula

Abstract We prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f, a pair $$h_1$$ h 1 , $$h_2$$ h 2 of zonal harmonic functions such that $$f=h_1-\bar{x} h_2$$ f = h 1 - x ¯ h 2 . We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.

scholarly journals On the Gauss mean-value formula for class number

1998 ◽  
Vol 151 ◽  
pp. 199-208 ◽  
Author(s):  
Fernando Chamizo ◽  
Henryk Iwaniec
Keyword(s):  
Class Number ◽  
Approximate Formula ◽  
Error Term ◽  
Exponential Sums ◽  
Mean Value ◽  
Point Problem ◽  
Class Number Formula ◽  
Modern Notation ◽  
Number Formula ◽  
Mean Value Formula

Abstract.In his masterwork Disquisitiones Arithmeticae, Gauss stated an approximate formula for the average of the class number for negative discriminants. In this paper we improve the known estimates for the error term in Gauss approximate formula. Namely, our result can be written as for every ∊ > 0, where H(−n) is, in modern notation, h(−4n). We also consider the average of h(−n) itself obtaining the same type of result.Proving this formula we transform firstly the problem in a lattice point problem (as probably Gauss did) and we use a functional equation due to Shintani and Dirichlet class number formula to express the error term as a sum of character and exponential sums that can be estimated with techniques introduced in a previous work on the sphere problem.

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