1<!doctype html> 2<html lang="en"> 3 <head> 4 <meta charset="utf-8"> 5 <title>Homepage</title> 6 <meta name="author" content="Bernd C. Kellner"> 7 <meta name="description" content="Number theory, Bernoulli numbers, Riemann zeta function, power sum"> 8 <meta name="keywords" content="Number theory, Bernoulli numbers, Riemann zeta function, power sum"> 9 <meta name="generator" content="HandWritten V1.0"/> 10 <meta name="viewport" content="width=device-width, initial-scale=1" /> 11 <meta name="format-detection" content="telephone=no"> 12 <link rel="icon" href="img/favicon.ico" type="image/x-icon"> 13 <link rel="stylesheet" href="css/w3.css"> 14 <link rel="stylesheet" href="css/style.css"> 15 </head> 16 <body class="nos"> 17 <header> 18 <div class="w3-top"> 19 <div class="w3-bar w3-blue-gray w3-padding w3-card" style="letter-spacing:2px;"> 20 <a href="#sect-home" class="w3-bar-item w3-button w3-round-large w3-hover-teal">Home</a> 21 <div class="w3-right w3-hide-small"> 22 <a href="#sect-publ" class="w3-bar-item w3-button w3-round-large w3-hover-teal">Publications</a> 23 <a href="#sect-form" class="w3-bar-item w3-button w3-round-large w3-hover-teal">Formulas</a> 24 <a href="#sect-link" class="w3-bar-item w3-button w3-round-large w3-hover-teal">Links</a> 25 </div> 26 </div> 27 </div> 28 </header> 29 30 <section id="sect-home"> 31 <h1>Homepage</h1> 32 Bernd C. Kellner<br> 33 Göttingen, Germany<br> 34 Email: bk (at) bernoulli.org<br><br> 35 </section> 36 37 <section> 38 <h1>Thesis</h1> 39 <cite>Ãber irreguläre Paare höherer Ordnungen</cite>. <br> 40 Diplomarbeit. Mathematisches Institut der Georg-August-Universität zu Göttingen, Germany, 2002.<br> 41 Online: <a class="tooltip-top" href="irrpairord.pdf">irrpairord.pdf<span class="tooltip-text">Download</span></a> (891 KB)<br><br> 42 </section> 43 44 <section id="sect-publ"> 45 <h1>Publications</h1> 46 <ol> 47 <li> 48 <cite>On irregular prime power divisors of the Bernoulli numbers</cite>, 49 Math. Comp. <b>76</b> (2007), <span class="cite-page">405–441</span>.<br> 50 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1183.11012" rel="noopener">1183.11012</a></span>   51 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.1090/S0025-5718-06-01887-4" rel="noopener">10.1090/S0025-5718-06-01887-4</a></span> 52 </li> 53 <li> 54 <cite>On asymptotic constants related to products of Bernoulli numbers and factorials</cite>, 55 Integers <b>9</b> (2009), Article A08, <span class="cite-page">83–106</span>.<br> 56 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1163.11014" rel="noopener">1163.11014</a></span>   57 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.1515/INTEG.2009.009" rel="noopener">10.1515/INTEG.2009.009</a></span>   58 <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol9.html" rel="noopener">Integers Vol. 9</a></span> 59 </li> 60 <li> 61 <cite>On stronger conjectures that imply the ErdÅs–Moser conjecture</cite>, 62 J. Number Theory <b>131</b> (2011), <span class="cite-page">1054–1061</span>.<br> 63 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1267.11031" rel="noopener">1267.11031</a></span>  
64 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.1016/j.jnt.2011.01.004" rel="noopener">10.1016/j.jnt.2011.01.004</a></span> 65 </li> 66 <li> 67 <cite>On quotients of Riemann zeta values at odd and even integer arguments</cite>, 68 J. Number Theory <b>133</b> (2013), <span class="cite-page">2684–2698</span>.<br> 69 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1290.11118" rel="noopener">1290.11118</a></span>   70 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.1016/j.jnt.2013.02.008" rel="noopener">10.1016/j.jnt.2013.02.008</a></span> 71 </li> 72 <li> 73 <cite>Identities between polynomials related to Stirling and harmonic numbers</cite>, 74 Integers <b>14</b> (2014), Article A54, <span class="cite-page">1–22</span>.<br> 75 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1315.11018" rel="noopener">1315.11018</a></span>   76 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.10452538" rel="noopener">10.5281/zenodo.10452538</a></span>   77 <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol14.html" rel="noopener">Integers Vol. 14</a></span> 78 </li> 79 <li> 80 <cite>The topology of Stein fillable manifolds in high dimensions II (with an appendix by <b>Bernd C. Kellner</b>)</cite>, 81 Geom. Topol. <b>19</b> (2015), <span class="cite-page">2995–3030</span>.<br> 82 Authors: <b>Jonathan Bowden</b>, <b>Diarmuid Crowley</b>, <b>András I. Stipsicz</b>. Appendix by <b>Bernd C. Kellner</b>.<br> 83 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1380.32016" rel="noopener">1380.32016</a></span>   84 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.2140/gt.2015.19.2995" rel="noopener">10.2140/gt.2015.19.2995</a></span> 85 </li> 86 <li> 87 <cite>On a product of certain primes</cite>, 88 J. Number Theory <b>179</b> (2017), <span class="cite-page">126–141</span>.<br> 89 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1418.11045" rel="noopener">1418.11045</a></span>   90 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.1016/j.jnt.2017.03.020" rel="noopener">10.1016/j.jnt.2017.03.020</a></span> 91 </li> 92 <li> 93 <cite>Power-sum denominators</cite>, 94 Amer. Math. Monthly <b>124</b> (2017), <span class="cite-page">695–709</span>.<br> 95 Co-author: <b>Jonathan Sondow</b><br> 96 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1391.11052" rel="noopener">1391.11052</a></span>   97 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.4169/amer.math.monthly.124.8.695" rel="noopener">10.4169/amer.math.monthly.124.8.695</a></span> 98 </li> 99 <li> 100 <cite>The denominators of power sums of arithmetic progressions</cite>, 101 Integers <b>18</b> (2018), Article A95, <span class="cite-page">1–17</span>.<br> 102 Co-author: <b>Jonathan Sondow</b><br> 103 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1423.11029" rel="noopener">1423.11029</a></span>  
104 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.10682734" rel="noopener">10.5281/zenodo.10682734</a></span>   105 <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol18.html" rel="noopener">Integers Vol. 18</a></span> 106 </li> 107 <li> 108 <cite>On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</cite>, 109 Integers <b>21</b> (2021), Article A52, <span class="cite-page">1–21</span>.<br> 110 Co-author: <b>Jonathan Sondow</b><br> 111 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1479.11028" rel="noopener">1479.11028</a></span>   112 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.10816833" rel="noopener">10.5281/zenodo.10816833</a></span>   113 <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol21.html" rel="noopener">Integers Vol. 21</a></span> 114 </li> 115 <li> 116 <cite>On (self-) reciprocal Appell polynomials: symmetry and Faulhaber-type polynomials</cite>, 117 Integers <b>21</b> (2021), Article A119, <span class="cite-page">1–19</span>.<br> 118 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1490.11033" rel="noopener">1490.11033</a></span>   119 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.10842328" rel="noopener">10.5281/zenodo.10842328</a></span>   120 <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol21.html" rel="noopener">Integers Vol. 21</a></span> 121 </li> 122 <li> 123 <cite>Shifted sums of the Bernoulli numbers, reciprocity, and denominators</cite>, 124 Rend. Mat. Appl. VII. Ser. <b>43</b> (2022), <span class="cite-page">151–163</span>.<br> 125 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1503.11047" rel="noopener">1503.11047</a></span>   126 <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="https://www1.mat.uniroma1.it/ricerca/rendiconti/43_2_(2022)_151-163.html" rel="noopener">Rend. Mat. Appl. VII. Ser. Vol. 43</a></span> 127 </li> 128 <li> 129 <cite>On primary Carmichael numbers</cite>, 130 Integers <b>22</b> (2022), Article A38, <span class="cite-page">1–39</span>.<br> 131 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1495.11008" rel="noopener">1495.11008</a></span>   132 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.10963985" rel="noopener">10.5281/zenodo.10963985</a></span>   133 <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol22.html" rel="noopener">Integers Vol. 22</a></span> 134 </li> 135 <li> 136 <cite>Faulhaber polynomials and reciprocal Bernoulli polynomials</cite>, 137 Rocky Mountain J. Math. <b>53</b> (2023), <span class="cite-page">119–151</span>.<br> 138 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1529.11039" rel="noopener">1529.11039</a></span>   139 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.1216/rmj.2023.53.119" rel="noopener">10.1216/rmj.2023.53.119</a></span> 140 </li> 141 <li> 142 <cite>On the nonintegrality of certain generalized binomial sums</cite>, 143 Integers <b>23</b> (2023), Article A45, <span class="cite-page">1–24</span>.<br>
144 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1548.11050" rel="noopener">1548.11050</a></span>   145 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.8099775" rel="noopener">10.5281/zenodo.8099775</a></span>   146 <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol23.html" rel="noopener">Integers Vol. 23</a></span> 147 </li> 148 <li> 149 <cite>On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients</cite>, 150 J. Integer Seq. <b>27</b> (2024), Article 24.2.8, <span class="cite-page">1–11</span>.<br> 151 <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1541.11023" rel="noopener">1541.11023</a></span>   152 <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Kellner/kell2.html" rel="noopener">Integer Seq. Vol. 27</a></span> 153 </li> 154 <li> 155 <cite>Asymptotic products of binomial and multinomial coefficients revisited</cite>, 156 Integers <b>24</b> (2024), Article A59, <span class="cite-page">1–10</span>.<br> 157 <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.12167556" rel="noopener">10.5281/zenodo.12167556</a></span>   158 <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol24.html" rel="noopener">Integers Vol. 24</a></span> 159 </li> 160 </ol> 161 </section> 162 163 <section> 164 <h1>Preprints</h1> 165 <ol type="A"> 166 <li> 167 <cite>The equation <span class="math"><span class="math-oper">denom</span>(<span class="math-var">B<sub class="math-var">n</sub></span>) = <span class="math-var">n</span></span> has only one solution</cite>. 168 Online: <a class="tooltip-top" href="denombneqn.pdf">denombneqn.pdf<span class="tooltip-text">Download</span></a> 169 </li> 170 <li> 171 <cite>On the <span class="math math-var">p</span>-divisibility of the sequence 172 <span class="math"><span class="math-var">B</span><sub class="math-var">lp<sup>r</sup></sub> / <span class="math-var">lp</span><sup class="math-var">r</sup></span></cite>. 173 Online: <a class="tooltip-top" href="pdivbn.pdf">pdivbn.pdf<span class="tooltip-text">Download</span></a> 174 </li> 175 <li> 176 <cite>Distribution modulo one and denominators of the Bernoulli polynomials</cite>, 177 <span class="cite-link"><b>arXiv</b>: <a class="w3-hover-shadow" target="_blank" href="https://arxiv.org/abs/1708.07119" rel="noopener">1708.07119</a></span>.<br> 178 </li> 179 <li> 180 My preprints on <a class="w3-hover-shadow" target="_blank" href="https://arxiv.org/find/math/1/au:+Kellner_B/0/1/0/all/0/1" rel="noopener"><b>arXiv.org</b></a> 181 </li> 182 </ol> 183 </section> 184 185 <section id="sect-form"> 186 <h1>Formulas</h1> 187 <h2 class="w3-round-large">Asymptotic products</h2> 188 <h3>Product of values of the gamma function</h3> 189 <p>Product of increasing powers of gamma values at rational arguments [<span class="cite-ref">2, Cor. 18, p. 93</span>]:</p> 190 <p class="w3-center"><img src="img/gamma1.png" width="362" height="81" alt="gamma1"></p> 191 <p>with the constants</p> 192 <p class="w3-center"><img src="img/gamma2.png" width="268" height="129" alt="gamma2"></p> 193 <p>where <span class="math">γ</span> is Euler's constant [<span class="cite-oeis">OEIS <a class="w3-hover-shadow " target="_blank" href="https://www.oeis.org/A001620" rel="noopener">A001620</a></span>] 194 and <span class="math"><img class="math-base" src="img/cca.png" width="14" height="14" alt="𝒜"></span> is the Glaisher-Kinkelin constant [<span class="cite-oeis">OEIS <a class="w3-hover-shadow" target="_blank" href="https://www.oeis.org/A074962" rel="noopener">A074962</a></span>].</p> 195 196 <h3>Product of factorials</h3> 197 <p>Special product ([<span class="cite-ref">2, Thms. 12, 13, p. 88</span>] for <span class="math"><span class="math-var">k</span> = <span class="math-num">5</span></span>):</p> 198 <p class="w3-center"><img src="img/factorial1.png" width="464" height="64" alt="factorial1"></p> 199 <p>with the constant</p> 200 <p class="w3-center"><img src="img/factorial2.png" width="453" height="101" alt="factorial2"></p> 201 <p>where <span class="math">(<span class="math-num">1</span>+√<span class="math-num">5</span>)/<span class="math-num">2</span></span> is the golden ratio [<span class="cite-oeis">OEIS <a class="w3-hover-shadow" target="_blank" href="https://www.oeis.org/A001622" rel="noopener">
201A001622</a></span>].</p> 202 203 <h3>Product of the Bernoulli numbers</h3> 204 <p>Product of (divided) Bernoulli numbers [<span class="cite-ref">2, Thm. 21, p. 95</span>]:</p> 205 <p class="w3-center"><img src="img/bnprod1.png" width="417" height="145" alt="bnprod1"></p> 206 <p>with the constants</p> 207 <p class="w3-center"><img src="img/bnprod2.png" width="360" height="141" alt="bnprod2"></p> 208 <p>where <span class="math"><img class="math-base" src="img/ccz.png" width="13" height="12" alt="𝒵"></span> is the product over all Riemann zeta values at even positive integer arguments [<span class="cite-oeis">OEIS <a class="w3-hover-shadow" target="_blank" href="https://www.oeis.org/A080729" rel="noopener">A080729</a></span>].</p> 209 210 <h2 class="w3-round-large">Number theory</h2> 211 <h3>Divisor function</h3> 212 <p>Recurrence formula for the divisor function for even <span class="math"><span class="math-var">n</span> ≥ <span class="math-num">8</span></span> [<span class="cite-ref">Thesis, Satz 1.1.5, p. 8</span>]:</p> 213 <p class="w3-center"><img src="img/divisor1.png" width="756" height="86" alt="divisor1"></p> 214 <p>with the convolution function</p> 215 <p class="w3-center"><img src="img/divisor2.png" width="281" height="67" alt="divisor2"></p> 216 <p>The first case <span class="math"><span class="math-var">n</span> = <span class="math-num">8</span></span> is known as Hurwitz's identity (1881)</p> 217 <p class="w3-center"><img src="img/divisor3.png" width="192" height="24" alt="divisor3"></p> 218 219 <h3>Identities with Stirling and harmonic numbers</h3> 220 <p>Define the polynomials</p> 221 <p class="w3-center"><img src="img/identsh1.png" width="568" height="65" alt="identsh1"></p> 222 <p>composed of Stirling numbers of the second kind <span class="math"><span class="math-bold">S</span><sub class="math-num">2</sub>(<span class="math-var">n</span>,<span class="math-var">k</span>)</span> and harmonic numbers <span class="math"><span class="math-bold">H</span><sub class="math-var">k</sub></span> by</p> 223 <p class="w3-center"><img src="img/identsh2.png" width="333" height="68" alt="identsh2"></p> 224 <p>The Genocchi numbers <span class="math"><span class="math-var">G</span><sub class="math-var">n</sub></span> [<span class="cite-oeis">OEIS <a class="w3-hover-shadow" target="_blank" href="https://www.oeis.org/A036968" rel="noopener">A036968</a></span>] are related to the Bernoulli numbers by</p> 225 <p class="w3-center"><img src="img/identsh3.png" width="280" height="24" alt="identsh3"></p> 226 <p>The second identity of <span class="math"><span class="math-var">B</span><sub class="math-var">n</sub></span> is due to Worpitzky (1883):</p> 227 <p class="w3-center"><img src="img/identsh4.png" width="367" height="65" alt="identsh4"></p> 228 <p>The <q>central value</q> is known to be</p> 229 <p class="w3-center"><img src="img/identsh5.png" width="162" height="51" alt="identsh5"></p> 230 <p>Involving the harmonic numbers leads to the identities [<span class="cite-ref">5, Thm. 1.3, p. 3</span>]:</p> 231 <p class="w3-center"><img src="img/identsh6.png" width="465" height="65" alt="identsh6"></p> 232 <p>and</p> 233 <p class="w3-center"><img src="img/identsh7.png" width="545" height="120" alt="identsh7"></p> 234 235 <h3>Riemann zeta function</h3> 236 <p>Quotient of Riemann zeta values [<span class="cite-ref">4, Thm. 1.3, p. 2687</span>]: If <span class="math"><span class="math-var">n</span> ≥ <span class="math-num">2</span></span> is even, then</p> 237 <p class="w3-center"><img src="img/zetaquot1.png" width="458" height="58" alt="zetaquot1"></p> 238 <p>with</p> 239 <p class="w3-center"><img src="img/zetaquot2.png" width="226" height="84" alt="zetaquot2"></p> 240 <p>where <span class="math"><img class="math-base" src="img/cclstar.png" width="18" height="13" alt="ℒ⋆"></span> is a linear functional, which is connected with a special Dirichlet series, 241 and <span class="math"><img src="img/lfpn.png" width="16" height="13" alt="𝔭n"></span> is a certain monic polynomial of degree <span class="math math-var">n</span>. 242 Moreover, if <span class="math"><span class="math-var">n</span> = <span class="math-var">p</span> + <span class="math-num">1</span></span> with <span class="math math-var">p</span> an odd prime, 243 then <span class="math"><img src="img/lfqn.png" width="16" height="13" alt="𝔮n"></span> is an Eisenstein polynomial and therefore irreducible over <span class="math">ℤ[<span class="math-var">x</span>]</span>.</p> 244 245 <h3>Conjecture on the structure of the Bernoulli numbers</h3> 246 <p>Under the assumption that every <span class="math math-var">p</span>-adic zeta function
247 <span class="math"><span class="math-var">ζ</span><sub>(<span class="math-var">p</span>,<span class="math-var">ℓ</span>)</sub>(<span class="math-var">s</span>)</span> has a unique simple zero 248 <span class="math"><span class="math-var">ξ</span><sub>(<span class="math-var">p</span>,<span class="math-var">ℓ</span>)</sub></span> 249 in case <span class="math">(<span class="math-var">p</span>,<span class="math-var">ℓ</span>)</span> is an irregular pair, 250 one has for even <span class="math"><span class="math-var">n</span> ≥ <span class="math-num">2</span></span> [<span class="cite-ref">1, Thm. 4.9, p. 16</span>]:</p> 251 <p class="w3-center"><img src="img/bnstruct1.png" width="518" height="88" alt="bnstruct1"></p> 252 <p>where</p> 253 <p class="w3-center"><img src="img/bnstruct2.png" width="415" height="133" alt="bnstruct2"></p> 254 <p>The denominator can be described by poles (always lying at <span class="math math-num">0</span>) and the numerator by zeros of <span class="math math-var">p</span>-adic zeta functions. Equivalently, the formula reads for the Bernoulli numbers:</p> 255 <p class="w3-center"><img src="img/bnstruct3.png" width="653" height="80" alt="bnstruct3"></p> 256 <p>Moreover, the formulas are valid for all irregular pairs <span class="math">(<span class="math-var">p</span>,<span class="math-var">ℓ</span>)</span> with</p> 257 <p class="w3-center"><img src="img/bnstruct4.png" width="234" height="25" alt="bnstruct4"></p> 258 259 <h3>Denominators of the Bernoulli polynomials</h3> 260 <p>The denominator of <span class="math"><span class="math-var">B</span><sub class="math-var">n</sub>(<span class="math-var">x</span>) − <span class="math-var">B</span><sub class="math-var">n</sub></span>, 261 the <span class="math math-var">n</span>th Bernoulli polynomial without constant term, is given by the remarkable formula ([<span class="cite-ref">7</span>], [<span class="cite-ref">8</span>] with Jonathan Sondow, [<span class="cite-oeis">OEIS <a class="w3-hover-shadow" target="_blank" href="https://www.oeis.org/A195441" rel="noopener">A195441</a></span>])</p> 262 <p class="w3-center"><img src="img/bpdenom1.png" width="311" height="56" alt="bpdenom1"></p> 263 <p>where <span class="math"><span class="math-var">s</span><sub class="math-var">p</sub>(<span class="math-var">n</span>)</span> denotes the sum of the base-<span class="math math-var">p</span> digits of <span class="math math-var">n</span>. 264 The finite product can be written with explicit sharp bounds as</p> 265 <p class="w3-center"><img src="img/bpdenom2.png" width="311" height="83" alt="bpdenom2"></p> 266 <p>where</p> 267 <p class="w3-center"><img src="img/bpdenom3.png" width="219" height="58" alt="bpdenom3"></p> 268 <p>The denominator of the <span class="math math-var">n</span>th Bernoulli polynomial <span class="math"><span class="math-var">B</span><sub class="math-var">n</sub>(<span class="math-var">x</span>)</span> can be described by a similar formula ([<span class="cite-ref">9</span>] with Jonathan Sondow):</p> 269 <p class="w3-center"><img src="img/bpdenom5.png" width="372" height="74" alt="bpdenom5"></p> 270 <p>If <span class="math">{<span style="vertical-align: middle">·</span>}</span> denotes the fractional part and <span class="math"><span class="math-var">n</span> ≥ <span class="math-num">1</span></span> is a fixed integer, 271 then there is the surprising relation [<span class="cite-ref">C</span>]:</p> 272 <p class="w3-center"><img src="img/bpdenom4.png" width="495" height="65" alt="bpdenom4"></p> 273 </section> 274 275 <section id="sect-link"> 276 <h1>Links</h1> 277 <ul> 278 <li><b>B. C. Kellner</b>: <a class="w3-hover-shadow" target="_blank" href="https://www.bernoulli.org" rel="noopener">The Bernoulli number page</a></li> 279 <li><b>K. Matthews</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.numbertheory.org" rel="noopener">Number Theory Web</a></li> 280 <li><b>OEIS</b>: <a class="w3-hover-shadow" target="_blank" href="https://oeis.org" rel="noopener">The On-Line Encyclopedia of Integer Sequences</a></li> 281 <li><b>
281Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://www.zbmath.org" rel="noopener">zbMATH Open</a></li> 282 </ul> 283 </section> 284 285 <footer class="w3-padding"> 286 <div class="w3-center"> 287 © Bernd C. Kellner, Göttingen, Germany, 2025.<br> 288 Email: bk (at) bernoulli.org<br> 289 <a class="w3-hover-shadow" style="font-size: smaller;" target="_blank" href="https://www.w3schools.com/w3css" rel="noopener">Powered by W3.CSS</a> 290 </div> 291 </footer> 292 </body> 293</html>
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