{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:00:39Z","timestamp":1776787239015,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"261","license":[{"start":{"date-parts":[[2008,5,14]],"date-time":"2008-05-14T00:00:00Z","timestamp":1210723200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    For any integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m greater-than 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">m&gt;1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    fix\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"zeta Subscript m Baseline equals exp left-parenthesis 2 pi i slash m right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b6\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>m<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext>exp<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\zeta _{m}=\\textrm {exp}(2 \\pi i\/m)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper Z Subscript m Superscript asterisk\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext mathvariant=\"bold\">Z<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>m<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\n                                \u2217\n                                \n                              <\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\textbf {Z}_{m}^{*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    denote the group of reduced residues modulo\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q equals p Superscript alpha\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03b1\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">q=p^{\\alpha }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , a power of a prime\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The hyper-\n                    <italic>Kloosterman sums of dimension<\/italic>\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are defined for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q\">\n                        <mml:semantics>\n                          <mml:mi>q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    by\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartLayout 1st Row  upper R left-parenthesis d comma q right-parenthesis equals sigma-summation Underscript x 1 comma period period period comma x Subscript n Baseline element-of upper Z Subscript q Superscript asterisk Baseline Endscripts zeta Subscript q Superscript x 1 plus midline-horizontal-ellipsis plus x Super Subscript n Superscript plus d left-parenthesis x 1 midline-horizontal-ellipsis x Super Subscript n Superscript right-parenthesis Super Superscript negative 1 Superscript Baseline left-parenthesis d element-of bold upper Z Subscript q Superscript asterisk Baseline right-parenthesis comma EndLayout\">\n                        <mml:semantics>\n                          <mml:mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\">\n                            <mml:mtr>\n                              <mml:mtd>\n                                <mml:mi>R<\/mml:mi>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>d<\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>q<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:munder>\n                                  <mml:mo>\n                                    \u2211\n                                    \n                                  <\/mml:mo>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:msub>\n                                      <mml:mi>x<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mn>1<\/mml:mn>\n                                      <\/mml:mrow>\n                                    <\/mml:msub>\n                                    <mml:mo>,<\/mml:mo>\n                                    <mml:mo>.<\/mml:mo>\n                                    <mml:mo>.<\/mml:mo>\n                                    <mml:mo>.<\/mml:mo>\n                                    <mml:mo>,<\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>x<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>n<\/mml:mi>\n                                      <\/mml:mrow>\n                                    <\/mml:msub>\n                                    <mml:mo>\n                                      \u2208\n                                      \n                                    <\/mml:mo>\n                                    <mml:msubsup>\n                                      <mml:mi>Z<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>q<\/mml:mi>\n                                      <\/mml:mrow>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mo>\n                                          \u2217\n                                          \n                                        <\/mml:mo>\n                                      <\/mml:mrow>\n                                    <\/mml:msubsup>\n                                  <\/mml:mrow>\n                                <\/mml:munder>\n                                <mml:msubsup>\n                                  <mml:mi>\n                                    \u03b6\n                                    \n                                  <\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>q<\/mml:mi>\n                                  <\/mml:mrow>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:msub>\n                                      <mml:mi>x<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mn>1<\/mml:mn>\n                                      <\/mml:mrow>\n                                    <\/mml:msub>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:mo>\n                                      \u22ef\n                                      \n                                    <\/mml:mo>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>x<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>n<\/mml:mi>\n                                      <\/mml:mrow>\n                                    <\/mml:msub>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:mi>d<\/mml:mi>\n                                    <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>x<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mn>1<\/mml:mn>\n                                      <\/mml:mrow>\n                                    <\/mml:msub>\n                                    <mml:mo>\n                                      \u22ef\n                                      \n                                    <\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>x<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>n<\/mml:mi>\n                                      <\/mml:mrow>\n                                    <\/mml:msub>\n                                    <mml:msup>\n                                      <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mo>\n                                          \u2212\n                                          \n                                        <\/mml:mo>\n                                        <mml:mn>1<\/mml:mn>\n                                      <\/mml:mrow>\n                                    <\/mml:msup>\n                                  <\/mml:mrow>\n                                <\/mml:msubsup>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>d<\/mml:mi>\n                                <mml:mo>\n                                  \u2208\n                                  \n                                <\/mml:mo>\n                                <mml:msubsup>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mtext mathvariant=\"bold\">Z<\/mml:mtext>\n                                  <\/mml:mrow>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>q<\/mml:mi>\n                                  <\/mml:mrow>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mo>\n                                      \u2217\n                                      \n                                    <\/mml:mo>\n                                  <\/mml:mrow>\n                                <\/mml:msubsup>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:mo>,<\/mml:mo>\n                              <\/mml:mtd>\n                            <\/mml:mtr>\n                          <\/mml:mtable>\n                          <mml:annotation encoding=\"application\/x-tex\">\\displaylines { R(d,q)= \\sum _{x_{1}, ..., x_{n} \\in Z_{q}^{*}} \\zeta _{q}^{x_{1}+ \\cdots +x_{n} +d(x_{1} \\cdots x_{n})^{-1}} \\;\\;\\;\\;\\;\\; (d \\in \\textbf {Z}_{q}^{*}), }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x Superscript negative 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\n                                \u2212\n                                \n                              <\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">x^{-1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    denotes the multiplicative inverse of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n                        <mml:semantics>\n                          <mml:mi>x<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    modulo\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q\">\n                        <mml:semantics>\n                          <mml:mi>q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Salie evaluated\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper R left-parenthesis d comma q right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">R(d,q)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the classical setting\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n=1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for even\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q\">\n                        <mml:semantics>\n                          <mml:mi>q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and for odd\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q equals p Superscript alpha\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03b1\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">q=p^{\\alpha }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha greater-than 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha &gt;1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Later, Smith provided formulas that simplified the computation of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper R left-parenthesis d comma q right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">R(d,q)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in these cases for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n greater-than 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n&gt;1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Recently, Cochrane, Liu and Zheng computed upper bounds for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper R left-parenthesis d comma q right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">R(d,q)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the general case\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n &gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , stopping short of their explicit evaluation. Here I complete the computation they initiated to obtain explicit values for the Kloosterman sums for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha greater-than 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha &gt;1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , relying on basic properties of some simple specialized exponential sums. The treatment here is more elementary than the author\u2019s previous determination of these Kloosterman sums using character theory and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -adic methods. At the least, it provides an alternative, independent evaluation of the Kloosterman sums.\n                  <\/p>","DOI":"10.1090\/s0025-5718-07-02011-x","type":"journal-article","created":{"date-parts":[[2007,10,29]],"date-time":"2007-10-29T06:30:33Z","timestamp":1193639433000},"page":"475-493","source":"Crossref","is-referenced-by-count":2,"title":["Explicit values of multi-dimensional Kloosterman sums for prime powers, II"],"prefix":"10.1090","volume":"77","author":[{"given":"S.","family":"Gurak","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2007,5,14]]},"reference":[{"key":"1","series-title":"Canadian Mathematical Society Series of Monographs and Advanced Texts","isbn-type":"print","volume-title":"Gauss and Jacobi sums","author":"Berndt, Bruce C.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0471128074"},{"key":"2","series-title":"Pure and Applied Mathematics, Vol. 20","volume-title":"Number theory","author":"Borevich, A. I.","year":"1966"},{"key":"3","doi-asserted-by":"crossref","unstructured":"J. Bourgain, \u201cExponential sum estimates on subgroups of \ud835\udc4d_{\ud835\udc5e}*, \ud835\udc5e arbitrary,\u201d J. Analyse Math. 97 (2005), 317-355.","DOI":"10.1007\/BF02807410"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/s00039-006-0558-7","article-title":"Exponential sum estimates over subgroups and almost subgroups of \u2124_{\u211a}*, where \u211a is composite with few prime factors","volume":"16","author":"Bourgain, J.","year":"2006","journal-title":"Geom. Funct. 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