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Comp."],"abstract":"<p>\n                    We improve the lower bound for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper V left-parenthesis upper T right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>V<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">V(T)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the number of sign changes of the error term\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"psi left-parenthesis x right-parenthesis minus x\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c8\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\psi (x)-x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the Prime Number Theorem in the interval\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket 1 comma upper T right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[1,T]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for large\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T\">\n                        <mml:semantics>\n                          <mml:mi>T<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">T<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We show that\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"limit inf StartFraction upper V left-parenthesis upper T right-parenthesis Over log upper T EndFraction greater-than-or-equal-to StartFraction gamma 0 Over pi EndFraction plus one sixtieth comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:munder>\n                              <mml:mo movablelimits=\"true\" form=\"prefix\">lim\u2006inf<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>T<\/mml:mi>\n                                <mml:mo stretchy=\"false\">\n                                  \u2192\n                                  \n                                <\/mml:mo>\n                                <mml:mi mathvariant=\"normal\">\n                                  \u221e\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:mi>V<\/mml:mi>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>T<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mi>log<\/mml:mi>\n                                <mml:mo>\n                                  \u2061\n                                  \n                                <\/mml:mo>\n                                <mml:mi>T<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mfrac>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mfrac>\n                              <mml:msub>\n                                <mml:mi>\n                                  \u03b3\n                                  \n                                <\/mml:mi>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mn>0<\/mml:mn>\n                                <\/mml:mrow>\n                              <\/mml:msub>\n                              <mml:mi>\n                                \u03c0\n                                \n                              <\/mml:mi>\n                            <\/mml:mfrac>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mfrac>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mn>60<\/mml:mn>\n                            <\/mml:mfrac>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\liminf _{T\\to \\infty }\\frac {V(T)}{\\log T}\\geq \\frac {\\gamma _{0}}{\\pi }+\\frac {1}{60},<\/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=\"gamma 0 equals 14.13 ellipsis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b3\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>14.13<\/mml:mn>\n                            <mml:mo>\n                              \u2026\n                              \n                            <\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\gamma _{0}=14.13\\ldots<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the imaginary part of the lowest-lying non-trivial zero of the Riemann zeta-function. The result is based on a new density estimate for zeros of the associated\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -function, over\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4 dot 10 Superscript 21\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mo>\n                              \u22c5\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>10<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>21<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">4\\cdot 10^{21}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    times better than previously known estimates of this type.\n                  <\/p>","DOI":"10.1090\/mcom\/4053","type":"journal-article","created":{"date-parts":[[2024,12,3]],"date-time":"2024-12-03T12:36:40Z","timestamp":1733229400000},"page":"3083-3100","source":"Crossref","is-referenced-by-count":0,"title":["On the sign changes of \ud835\udf13(\ud835\udc65)-\ud835\udc65"],"prefix":"10.1090","volume":"94","author":[{"given":"Maciej","family":"Grze\u015bkowiak","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jerzy","family":"Kaczorowski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"\u0141ukasz","family":"Pa\u0144kowski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maciej","family":"Radziejewski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2025,1,27]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"691","DOI":"10.1112\/S0024610700008772","article-title":"The best quantitative Kronecker\u2019s theorem","volume":"61","author":"Chen, Yong-Gao","year":"2000","journal-title":"J. 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