cp's OEIS Frontend

A385495 Most significant nonzero decimal digit of zeta(n)-1, where zeta(n) = Sum_{j >= 1} 1/j^n is the Riemann zeta function.

%I A385495 #14 Jul 23 2025 19:10:54
%S A385495 6,2,8,3,1,8,4,2,9,4,2,1,6,3,1,7,3,1,9,4,2,1,5,2,1,7,3,1,9,4,2,1,5,2,
%T A385495 1,7,3,1,9,4,2,1,5,2,1,7,3,1,8,4,2,1,5,2,1,6,3,1,8,4,2,1,5,2,1,6,3,1,
%U A385495 8,4,2,1,5,2,1,6,3,1,8,4,2,1,5,2,1,6,3
%N A385495 Most significant nonzero decimal digit of zeta(n)-1, where zeta(n) = Sum_{j >= 1} 1/j^n is the Riemann zeta function.
%C A385495 The sequence starts at n = 2 since the zeta function sum diverges for any integer n < 2.
%C A385495 Conjecture: a(n) = A111395(n) for all n >= 10 (for a weaker conjecture see the comments in the related sequence A385431 and the Mathematics Stack Exchange discussion).
%D A385495 Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%H A385495 Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High Precision Computation of Hardy-Littlewood Constants</a>, Preprint, 1998.
%H A385495 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/5079681/is-the-leading-digit-of-the-decimal-expansion-of-the-prime-zeta-function-at-n">Is the leading digit of the decimal expansion of the prime zeta function at n equal to the first digit of 5^n, for all integers n >= 10?</a>.
%H A385495 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RiemannZetaFunction.html">Riemann Zeta Function</a>.
%H A385495 Wikipedia, <a href="https://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann Zeta function</a>.
%e A385495 For n = 4, a(4) = 8 since the most significant digit of zeta(4)-1 = 0.0823232... is 8 (see A013662).
%t A385495 Table[First@Select[First@RealDigits[N[Zeta[n] - 1, 100]], # != 0 &], {n, 2, 100}]
%Y A385495 Cf. A000027, A013662, A085548, A085541, A085964 to A085969, A111395, A385431.
%K A385495 nonn,easy,base
%O A385495 2,1
%A A385495 _Marco Ripà_, Jun 30 2025