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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| Published in | Journal of computational and applied mathematics Vol. 157; no. 2; pp. 297 - 308 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier B.V
15.08.2003
Elsevier |
| Subjects | |
| Online Access | Get full text |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. The relation between the exponential sums SN(x;p)=*Sn=0N-1exp(*pixnp) and T0=T0(x;N,p)=*Sn=1infinitye-n/Nexp(*pixNpe-pn /N), where x# > 0 and p > 0, is investigated. It is demonstrated that there is an asymptotic connection as NRTinfinity which is found numerically to be valid provided the variable x satisfies the restriction xNp=o(N) when p > 1. The sum T0 is shown to be associated with a zeta function defined by Z(s)=*Sn=1infinityexp(i*ce-an)n-s for real *c and a > 0. |
| Author | Paris, R.B. |
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| Cites_doi | 10.1090/S0273-0979-1981-14930-2 10.1088/0951-7715/1/1/001 10.1112/S0025579300008718 10.1007/BF02401833 |
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| Keywords | Exponential sums Asymptotics Curlicues Asymptotic behavior Zeta function Exponential sum Laplace integral Laplace transformation Asymptotic approximation Exponential function Summation |
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| References | Temme (BIB12) 1994 Whittaker, Watson (BIB14) 1952 Titchmarsh (BIB13) 1988 Dekking, Mendès-France (BIB5) 1981; 329 Andrews, Askey, Roy (BIB1) 1999 Berry, Goldberg (BIB3) 1988; 1 Hardy, Littlewood (BIB6) 1914; 37 Coutsias, Kazarinoff (BIB4) 1987; 26D Lehmer (BIB7) 1976; 23 R.B. Paris, On the growth of an exponential sum: a survey, Technical Report MS 02:01, University of Abertay Dundee, 2002. Lindelöf (BIB8) 1905 Berndt, Evans (BIB2) 1981; 5 Olver (BIB9) 1974 Paris, Kaminski (BIB10) 2001 Berndt (10.1016/S0377-0427(03)00412-6_BIB2) 1981; 5 Berry (10.1016/S0377-0427(03)00412-6_BIB3) 1988; 1 Coutsias (10.1016/S0377-0427(03)00412-6_BIB4) 1987; 26D Whittaker (10.1016/S0377-0427(03)00412-6_BIB14) 1952 Paris (10.1016/S0377-0427(03)00412-6_BIB10) 2001 Dekking (10.1016/S0377-0427(03)00412-6_BIB5) 1981; 329 Lindelöf (10.1016/S0377-0427(03)00412-6_BIB8) 1905 Temme (10.1016/S0377-0427(03)00412-6_BIB12) 1994 Titchmarsh (10.1016/S0377-0427(03)00412-6_BIB13) 1988 Andrews (10.1016/S0377-0427(03)00412-6_BIB1) 1999 Hardy (10.1016/S0377-0427(03)00412-6_BIB6) 1914; 37 Olver (10.1016/S0377-0427(03)00412-6_BIB9) 1974 10.1016/S0377-0427(03)00412-6_BIB11 Lehmer (10.1016/S0377-0427(03)00412-6_BIB7) 1976; 23 |
| References_xml | – volume: 26D start-page: 295 year: 1987 end-page: 310 ident: BIB4 article-title: Disorder, renormalizability, theta functions and Cornu spirals publication-title: Physica contributor: fullname: Kazarinoff – year: 1988 ident: BIB13 publication-title: The theory of the Riemann Zeta-Function contributor: fullname: Titchmarsh – year: 1999 ident: BIB1 publication-title: Special Functions contributor: fullname: Roy – year: 2001 ident: BIB10 publication-title: Asymptotics and Mellin–Barnes Integrals contributor: fullname: Kaminski – year: 1994 ident: BIB12 article-title: Bernoulli polynomials old and new: problems in analysis and asymptotics publication-title: From Universal Morphisms to Megabytes: A Baayen Space Odyssey contributor: fullname: Temme – volume: 5 start-page: 107 year: 1981 end-page: 129 ident: BIB2 article-title: The determination of Gauss sums publication-title: Bull. Amer. Math. Soc. contributor: fullname: Evans – volume: 37 start-page: 183 year: 1914 end-page: 236 ident: BIB6 article-title: Some problems of Diophantine approximation. II publication-title: Acta Math. contributor: fullname: Littlewood – year: 1952 ident: BIB14 publication-title: A Course of Modern Analysis contributor: fullname: Watson – year: 1905 ident: BIB8 publication-title: Le Calcul des Résidus contributor: fullname: Lindelöf – year: 1974 ident: BIB9 publication-title: Asymptotics and Special Functions contributor: fullname: Olver – volume: 1 start-page: 1 year: 1988 end-page: 26 ident: BIB3 article-title: Renormalisation of curlicues publication-title: Nonlinearity contributor: fullname: Goldberg – volume: 329 start-page: 143 year: 1981 end-page: 153 ident: BIB5 article-title: Uniform distribution modulo one publication-title: J. Reine Angew. Math. contributor: fullname: Mendès-France – volume: 23 start-page: 125 year: 1976 end-page: 135 ident: BIB7 article-title: Incomplete Gauss sums publication-title: Mathematika contributor: fullname: Lehmer – year: 1974 ident: 10.1016/S0377-0427(03)00412-6_BIB9 contributor: fullname: Olver – year: 1999 ident: 10.1016/S0377-0427(03)00412-6_BIB1 contributor: fullname: Andrews – year: 1988 ident: 10.1016/S0377-0427(03)00412-6_BIB13 contributor: fullname: Titchmarsh – volume: 5 start-page: 107 year: 1981 ident: 10.1016/S0377-0427(03)00412-6_BIB2 article-title: The determination of Gauss sums publication-title: Bull. Amer. Math. Soc. doi: 10.1090/S0273-0979-1981-14930-2 contributor: fullname: Berndt – volume: 329 start-page: 143 year: 1981 ident: 10.1016/S0377-0427(03)00412-6_BIB5 article-title: Uniform distribution modulo one publication-title: J. Reine Angew. Math. contributor: fullname: Dekking – year: 1952 ident: 10.1016/S0377-0427(03)00412-6_BIB14 contributor: fullname: Whittaker – volume: 1 start-page: 1 year: 1988 ident: 10.1016/S0377-0427(03)00412-6_BIB3 article-title: Renormalisation of curlicues publication-title: Nonlinearity doi: 10.1088/0951-7715/1/1/001 contributor: fullname: Berry – ident: 10.1016/S0377-0427(03)00412-6_BIB11 – volume: 26D start-page: 295 year: 1987 ident: 10.1016/S0377-0427(03)00412-6_BIB4 article-title: Disorder, renormalizability, theta functions and Cornu spirals publication-title: Physica contributor: fullname: Coutsias – volume: 23 start-page: 125 year: 1976 ident: 10.1016/S0377-0427(03)00412-6_BIB7 article-title: Incomplete Gauss sums publication-title: Mathematika doi: 10.1112/S0025579300008718 contributor: fullname: Lehmer – volume: 37 start-page: 183 year: 1914 ident: 10.1016/S0377-0427(03)00412-6_BIB6 article-title: Some problems of Diophantine approximation. II publication-title: Acta Math. doi: 10.1007/BF02401833 contributor: fullname: Hardy – year: 1994 ident: 10.1016/S0377-0427(03)00412-6_BIB12 article-title: Bernoulli polynomials old and new: problems in analysis and asymptotics contributor: fullname: Temme – year: 2001 ident: 10.1016/S0377-0427(03)00412-6_BIB10 contributor: fullname: Paris – year: 1905 ident: 10.1016/S0377-0427(03)00412-6_BIB8 contributor: fullname: Lindelöf |
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| Snippet | 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... The relation between the exponential sums SN(x;p)=*Sn=0N-1exp(*pixnp) and T0=T0(x;N,p)=*Sn=1infinitye-n/Nexp(*pixNpe-pn /N), where x# > 0 and p > 0, is... |
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| SubjectTerms | 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 |
| Title | On the asymptotic connection between two exponential sums |
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