singular series
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Author(s):  
C. FREI ◽  
P. KOYMANS ◽  
E. SOFOS

The first purpose of our paper is to show how Hooley’s celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy–Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series. In particular, using results of Lenstra, Stevenhagen and Moree, we provide a partial factorisation as an Euler product and prove that this does not extend to a complete factorisation.


2017 ◽  
Vol 56 (1) ◽  
pp. 117-141
Author(s):  
Daniel A. Goldston ◽  
Julian Ziegler Hunts ◽  
Timothy Ngotiaoco
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2014 ◽  
Vol 57 (3) ◽  
pp. 681-692
Author(s):  
JÖRG BRÜDERN

1. Let r(n) denote the number of representations of the natural number n as the sum of one square and three fifth powers of positive integers. A formal use of the circle method predicts the asymptotic relation (1)$ \begin{equation*} r(n) = \frac{\Gamma(\frac32)\Gamma(\frac65)^3}{\Gamma(\frac{11}{10})} {\mathfrak s}(n) {n}^\frac1{10} (1 + o(1)) \qquad (n\to\infty). \end{equation*} $ Here ${\mathfrak s}$(n) is the singular series associated with sums of a square and three fifth powers, see (13) below for a precise definition. The main purpose of this note is to confirm (1) in mean square.


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