Two-Dimensional Period Estimation by Ramanujan's Sum

2017 ◽  
Vol 65 (19) ◽  
pp. 5108-5120 ◽  
Author(s):  
Soo-Chang Pei ◽  
Kuo-Wei Chang
1986 ◽  
Vol 59 (4) ◽  
pp. 216 ◽  
Author(s):  
Kenneth R. Johnson

1979 ◽  
Vol 10 (1) ◽  
pp. 71-87 ◽  
Author(s):  
J. Chidambaraswamy

2011 ◽  
Vol 54 (1) ◽  
pp. 155-162 ◽  
Author(s):  
ZHANG WENPENG

AbstractLet q > 1 be an odd integer and c be a fixed integer with (c, q) = 1. For each integer a with 1 ≤ a ≤ q − 1, it is clear that there exists one and only one b with 0 ≤ b ≤ q − 1 such that ab ≡ c (mod q). Let N(c, q) denotes the number of all solutions of the congruence equation ab ≡ c (mod q) for 1 ≤ a, b ≤ q − 1 in which a and b are of opposite parity, where b is defined by the congruence equation bb ≡ 1(modq). The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to study the mean value properties of a summation involving (N(c, q) − φ(q)) and Ramanujan's sum, and give two exact computational formulae.


2012 ◽  
Vol 433-440 ◽  
pp. 2997-3002 ◽  
Author(s):  
Hairudin Abdul Majid ◽  
Lili Ayu Wulandhari ◽  
Azzurah A. Samah ◽  
Ang Jun Chin

A framework for determining optimal period and cost in extended warranty is proposed in this paper. The optimal period and cost is derived from two-dimensional delay time model. Two-dimensional delay time model is governed by age and mileage. Age and mileage gives effect for the rate of defect arrival λ. The steps for obtaining the model to estimate the optimal period cost of extended warranty is presented. It starts from modeling the defect arrival rate, the impact of preventive maintenance toward the defect arrival rate; determine the delay time distribution, data analyzing and the last optimal cost and period estimation for extended warranty.


1966 ◽  
Vol s1-41 (1) ◽  
pp. 595-604 ◽  
Author(s):  
M. V. Subba Rao ◽  
V. C. Harris

1980 ◽  
Vol 32 (5) ◽  
pp. 1250-1260 ◽  
Author(s):  
K. G. Ramanathan ◽  
M. V. Subbarao

Ramanujan's well known trigonometrical sum C(m, n) denned bywhere x runs through a reduced residue system (mod n), had been shown to occur in analytic problems concerning modular functions of one variable, by Poincaré [4]. Ramanujan, independently later, used these trigonometrical sums in his remarkable work on representation of integers as sums of squares [6]. There are various generalizations of C(m, n) in the literature (some also to algebraic number fields); see, for example, [9] which gives references to some of these. Perhaps the earliest generalization to algebraic number fields is due to H. Rademacher [5]. We here consider a novel generalization involving matrices.


2019 ◽  
Vol 16 (01) ◽  
pp. 65-76
Author(s):  
Yujie Wang ◽  
Chungang Ji

In this paper, we generalize Ramanujan’s sum to the ring of integers of an algebraic number field. We also obtain the orthogonality properties of Ramanujan’s sum in the ring of integers.


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