Comments - Two New Deep Conjectures in Probabilistic Number Theory - Data Science Central 2021-06-17T07:52:12Z https://www.datasciencecentral.com/profiles/comment/feed?attachedTo=6448529%3ABlogPost%3A884557&xn_auth=no Two remarks: In figure 2, do… tag:www.datasciencecentral.com,2020-01-22:6448529:Comment:924745 2020-01-22T07:04:49.977Z Vincent Granville https://www.datasciencecentral.com/profile/VincentGranville <p>Two remarks:</p> <ul> <li>In figure 2, does the curve cross the X-axis infinitely often?</li> <li>When approximating any number (say Pi) by fractions (ratios of integers) only the first <i>m</i> digits of the denominator <em>x</em>(<em>n</em>) should be studied to determine normality, where <em>m</em> is the integer part of the logarithm in base 2, of the denominator of <em>x</em>(<em>n</em>). More about this…</li> </ul> <p>Two remarks:</p> <ul> <li>In figure 2, does the curve cross the X-axis infinitely often?</li> <li>When approximating any number (say Pi) by fractions (ratios of integers) only the first <i>m</i> digits of the denominator <em>x</em>(<em>n</em>) should be studied to determine normality, where <em>m</em> is the integer part of the logarithm in base 2, of the denominator of <em>x</em>(<em>n</em>). More about this <a href="https://math.stackexchange.com/questions/3513257/conjecture-about-the-distribution-of-0-1-in-the-binary-expansion-of-rational-n/3518264#3518264" target="_blank" rel="noopener">here</a>.</li> <li>Let <em>u</em>(<em>n</em>) be the number of digits equal to zero, and let <em>v</em>(<em>n</em>) be the number of digits equal to one in the first <em>n</em> binary digits. If <em>u</em>(<em>n</em>) - <em>v</em>(<em>n</em>) = <em>O</em>(<em>n</em>^<span>α)</span>, with <span>α &lt; 1, </span>then the digit distribution is uniform. Typically, <span>α = 1/2.</span></li> </ul> <p></p>