On the asymptotic connection between two exponential sums

The relation between the exponential sums S N(x;p)=∑ n=0 N−1 exp(π ixn p) and T 0≡T 0(x;N,p)=∑ n=1 ∞ e −n/N exp(π ixN p e −pn/N) , where x⩾0 and p>0, is investigated. It is demonstrated that there is an asymptotic connection as N→∞ which is found numerically to be valid provided the variable x sa...

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Bibliographic Details
Published inJournal of computational and applied mathematics Vol. 157; no. 2; pp. 297 - 308
Main Author Paris, R.B.
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier B.V 15.08.2003
Elsevier
Subjects
Approximations and expansions
Asymptotics
Curlicues
Exact sciences and technology
Exponential sums
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Sciences and techniques of general use
Sequences, series, summability
Exponential sums
Asymptotics
Curlicues
Asymptotic behavior
Zeta function
Exponential sum
Laplace integral
Laplace transformation
Asymptotic approximation
Exponential function
Summation
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Summary:The relation between the exponential sums S N(x;p)=∑ n=0 N−1 exp(π ixn p) and T 0≡T 0(x;N,p)=∑ n=1 ∞ e −n/N exp(π ixN p e −pn/N) , where x⩾0 and p>0, is investigated. It is demonstrated that there is an asymptotic connection as N→∞ which is found numerically to be valid provided the variable x satisfies the restriction xN p =o( N) when p>1. The sum T 0 is shown to be associated with a zeta function defined by Z(s)=∑ n=1 ∞ exp( iθ e −an)n −s for real θ and a>0.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 23
ISSN:0377-0427
1879-1778
DOI:10.1016/S0377-0427(03)00412-6