scholarly journals The tail of the singular series for the prime pair and Goldbach problems

2017 ◽  
Vol 56 (1) ◽  
pp. 117-141
Author(s):  
Daniel A. Goldston ◽  
Julian Ziegler Hunts ◽  
Timothy Ngotiaoco
Keyword(s):  
Prime Pair ◽  
Singular Series

PRIME PAIR LABELING OF GRAPHS AND DIGRAPHS

2020 ◽  
Vol 9 (5) ◽  
pp. 2687-2700
Author(s):  
G. Suganya ◽  
K. Palani ◽  
T. M. Velammal
Keyword(s):  
Prime Pair

scholarly journals New Estimates of the Singular Series Corresponding to Positive Quaternary Quadratic Forms

2006 ◽  
Vol 13 (4) ◽  
pp. 687-691
Author(s):  
Guram Gogishvili
Keyword(s):  
Quadratic Form ◽  
General Result ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Precise Estimate ◽  
Definite Integral ◽  
Singular Series ◽  
Prime Factors ◽  
Quaternary Quadratic Form ◽  
Best Estimates

Abstract Let 𝑚 ∈ ℕ, 𝑓 be a positive definite, integral, primitive, quaternary quadratic form of the determinant 𝑑 and let ρ(𝑓,𝑚) be the corresponding singular series. When studying the best estimates for ρ(𝑓,𝑚) with respect to 𝑑 and 𝑚 we proved in [Gogishvili, Trudy Tbiliss. Univ. 346: 72–77, 2004] that where 𝑏(𝑘) is the product of distinct prime factors of 16𝑘 if 𝑘 ≠ 1 and 𝑏(𝑘) = 3 if 𝑘 = 1. The present paper proves a more precise estimate where 𝑑 = 𝑑0𝑑1, if 𝑝 > 2; 𝑕(2) ⩾ –4. The last estimate for ρ(𝑓,𝑚) as a general result for quaternary quadratic forms of the above-mentioned type is unimprovable in a certain sense.

Clitics, affixes, and the evolution of the question marker ‘tu’ in Canadian French

1991 ◽  
Vol 1 (2) ◽  
pp. 179-187 ◽  
Author(s):  
Marc Picard
Keyword(s):  
Singular Series ◽  
Canadian French ◽  
The Third ◽  
Third Person ◽  
History Of ◽  
The Third Person

AbstractClitics and affixes are known to originate from erstwhile independent words. In French, a question marker has developed in a most peculiar way through the combination of a verb-final consonant and the third person masculine singular pronoun, and has gradually spread to other persons through a singular series of phonological, syntactic and analogical processes. Although this it is now all but moribund in Continental Frenceh, its offshoot tu is alive and well in Canadian French, the construction subject + verb + tu having become the most usual way of formulating yes-no questions in this dialect. Despite the long history of ti\tu, however, its exact grammatical status has yet to be established. Various criteria that have been proposed in recent years to distinguish clitics from affixes would seem to indicate that this morpheme should be properly classified as a suffis.

On a Generalization of the Prime Pair Problem

1957 ◽  
Vol 11 (60) ◽  
pp. 249 ◽  
Author(s):  
Herschel F. Smith
Keyword(s):  
Prime Pair

scholarly journals On the summation of the singular series

1987 ◽  
Vol 57 (4) ◽  
pp. 469-475
Author(s):  
A. Arenas
Keyword(s):  
Singular Series
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