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A385431 Leading digit of the decimal expansion of the prime zeta function at n.

%I A385431 #25 Aug 04 2025 18:39:26
%S A385431 4,1,7,3,1,8,4,2,9,4,2,1,6,3,1,7,3,1,9,4,2,1,5,2,1,7,3,1,9,4,2,1,5,2,
%T A385431 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 A385431 8,4,2,1,5,2,1,6,3,1,8,4,2,1,5,2,1,6,3
%N A385431 Leading digit of the decimal expansion of the prime zeta function at n.
%C A385431 For each n = 2, 3, 4, ..., a(n) is the most significant (nonzero) digit of the decimal expansion of P(n) := Sum_{p prime} 1/p^n, the prime zeta function at argument n.
%C A385431 The present sequence starts at n = 2, since the underlying series diverges for any integer less than 2.
%C A385431 It is conjectured that a(n) = A111395(n) for all n >= 10 (see "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?" in Links).
%D A385431 Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%D A385431 J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.
%H A385431 Robert G. Wilson v, <a href="/A385431/b385431.txt">Table of n, a(n) for n = 2..1001</a>
%H A385431 Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High Precision Computation of Hardy-Littlewood Constants</a>, Preprint, 1998.
%H A385431 Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%H A385431 X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>.
%H A385431 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 A385431 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/4655237/what-is-the-sum-of-negative-integer-powers-of-all-prime-numbers">What is the sum of negative integer powers of all prime numbers?</a>.
%H A385431 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>.
%H A385431 Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Prime_zeta_function">Prime zeta function</a>.
%F A385431 a(n) = most significant (nonzero) digit of P(n), where P(n) := Sum_{p prime} 1/p^n.
%F A385431 a(n) = ld(P(n)), where ld(x) := floor(x/10^floor(log_10(x))) and P(n) := Sum_{k >= 1} moebius(k)*log(zeta(n*k))/k.
%F A385431 For all n > 9, a(n) = most significant (nonzero) digit of 5^n (conjectured).
%e A385431 For n = 4, a(4) = 7 since the most significant digit of P(4) = Sum_{p prime} 1/p^4 = 0.07699313976424684494... is 7.
%t A385431 Table[Module[{digits, firstNonZero}, digits = First[RealDigits[N[Sum[MoebiusMu[n]*Log[Zeta[k*n]]/n, {n, 1, 200}], 100]]]; firstNonZero = Select[digits, Function[d, d != 0]][[1]]; firstNonZero], {k, 2, 88}]
%t A385431 $MaxExtraPrecision = 2^10; a[n_] := RealDigits[ Sum[ MoebiusMu[m]*Log[ Zeta[n*m]]/m,{m, 32}], 10, 16][[1, 1]]; Array[a, 87, 2] - (* _Robert G. Wilson v_, Jul 11 2025 *)
%o A385431 (PARI) a(n) = my(x=sumeulerrat(1/p, n)); while(x<1, x*=10); floor(x); \\ _Michel Marcus_, Jun 29 2025
%Y A385431 Cf. A000040, A085548, A085541, A085964 to A085969, A111395.
%K A385431 nonn,easy,base
%O A385431 2,1
%A A385431 _Marco Ripà_, Jun 28 2025