Singular Series of Ascending–Descending Powers

2010 ◽  
pp. 415-428
Keyword(s):  
Singular Series

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.

scholarly journals On the summation of the singular series

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

Some Flint Tools of the Iron Age: a Singular Series

1925 ◽  
Vol 5 (2) ◽  
pp. 158-163 ◽  
Author(s):  
H. G. O. Kendall
Keyword(s):  
Iron Age ◽  
Singular Series ◽  
Know How

Some years ago Dr. Blackmore discovered, on top of Laverstock Down, a hitherto unknown series of flint tools, turned up by the plough, which he named ‘Rectangular’. He has pointed out that it is in their extreme simplicity, together with constancy to type, that the skill in making them lies; for the manufacturers of these tools chipped flakes of one pattern, with several particular features, by the hundred. Modern practitioners know how many blows have to be given to knock a flake or a block into a required shape, but these people made one kind of tool with some halfdozen strokes, almost unfailingly, over and over again.

scholarly journals Vinogradov’s three primes theorem with primes having given primitive roots

2019 ◽  
pp. 1-36
Author(s):  
C. FREI ◽  
P. KOYMANS ◽  
E. SOFOS
Keyword(s):  
Circle Method ◽  
Singular Series ◽  
Euler Product ◽  
Primitive Roots ◽  
Do So

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.

On Waring's problem for cubes

1991 ◽  
Vol 109 (2) ◽  
pp. 229-256 ◽  
Author(s):  
Jörg Brüdern
Keyword(s):  
Singular Series ◽  
Natural Numbers ◽  
Additive Theory ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Quantitative Form ◽  
Image Position ◽  
Theory Of Numbers

A classical conjecture in the additive theory of numbers is that all sufficiently large natural numbers may be written as the sum of four positive cubes of integers. This is known as the Four Cubes Problem, and since the pioneering work of Hardy and Littlewood one expects a much more precise quantitative form of the conjecture to hold. Let v(n) be the number of representations of n in the proposed manner. Then the expected formula takes the shapewhere (n) is the singular series associated with four cubes as familiar in the Hardy–Littlewood theory.

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