scholarly journals Mean square value of exponential sums related to the representation of integers as sums of squares

2005 ◽  
Vol 117 (4) ◽  
pp. 353-370 ◽  
Author(s):  
Jens Marklof
Keyword(s):  
Exponential Sums ◽  
Sums Of Squares ◽  
Mean Square ◽  
Representation Of Integers

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.

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