Ramanujan’s Sum and Its Generalizations

2018 ◽  
pp. 171-196
Author(s):  
R. Sivaramakrishnan
Keyword(s):  
Ramanujan’S Sum ◽  
Ramanujan's Sum

Reciprocity in Ramanujan's Sum

1986 ◽  
Vol 59 (4) ◽  
pp. 216 ◽  
Author(s):  
Kenneth R. Johnson
Keyword(s):  
Ramanujan’S Sum ◽  
Ramanujan's Sum

Generalized Ramanujan's sum

1979 ◽  
Vol 10 (1) ◽  
pp. 71-87 ◽  
Author(s):  
J. Chidambaraswamy
Keyword(s):  
Ramanujan’S Sum ◽  
Ramanujan's Sum

scholarly journals A MEAN VALUE RELATED TO D. H. LEHMER'S PROBLEM AND THE RAMANUJAN'S SUM

2011 ◽  
Vol 54 (1) ◽  
pp. 155-162 ◽  
Author(s):  
ZHANG WENPENG
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Opposite Parity ◽  
Fixed Integer ◽  
All Solutions ◽  
The Mean ◽  
Ramanujan’S Sum ◽  
Lehmer’S Problem ◽  
Congruence Equation ◽  
Ramanujan's Sum

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.

Two-Dimensional Period Estimation by Ramanujan's Sum

2017 ◽  
Vol 65 (19) ◽  
pp. 5108-5120 ◽  
Author(s):  
Soo-Chang Pei ◽  
Kuo-Wei Chang
Keyword(s):  
Two Dimensional ◽  
Period Estimation ◽  
Ramanujan’S Sum ◽  
Ramanujan's Sum

A New Generalization of Ramanujan's Sum

1966 ◽  
Vol s1-41 (1) ◽  
pp. 595-604 ◽  
Author(s):  
M. V. Subba Rao ◽  
V. C. Harris
Keyword(s):  
Ramanujan’S Sum ◽  
Ramanujan's Sum

Some Generalizations of Ramanujan's Sum

1980 ◽  
Vol 32 (5) ◽  
pp. 1250-1260 ◽  
Author(s):  
K. G. Ramanathan ◽  
M. V. Subbarao
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Sums Of Squares ◽  
Modular Functions ◽  
Algebraic Number Fields ◽  
Representation Of Integers ◽  
Ramanujan’S Sum ◽  
Image Position ◽  
Ramanujan's Sum

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.

Ramanujan’s sum in the ring of integers of an algebraic number field

2019 ◽  
Vol 16 (01) ◽  
pp. 65-76
Author(s):  
Yujie Wang ◽  
Chungang Ji
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Ring Of Integers ◽  
Ramanujan’S Sum ◽  
Ramanujan's Sum

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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