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

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, 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, 8, 4, 2, 1, 5, 2, 1, 6, 3, 1, 8, 4, 2, 1, 5, 2, 1, 6, 3

Offset: 2

Author

Marco Ripà, Jun 28 2025

Keywords

• nonn
• easy
• base

Comments

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.

The present sequence starts at n = 2, since the underlying series diverges for any integer less than 2.

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).

Examples

			For n = 4, a(4) = 7 since the most significant digit of P(4) = Sum_{p prime} 1/p^4 = 0.07699313976424684494... is 7.
		

References

• Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
• J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 2..1001
• Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.
• Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
• X. Gourdon and P. Sebah, Some Constants from Number theory.
• Mathematics Stack Exchange, 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?.
• Mathematics Stack Exchange, What is the sum of negative integer powers of all prime numbers?.
• Eric Weisstein's World of Mathematics, Prime Zeta Function.
• Wikipedia, Prime zeta function.

Crossrefs

Cf. A000040, A085548, A085541, A085964 to A085969, A111395.

Programs

Mathematica

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}]
$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 *)

PARI

a(n) = my(x=sumeulerrat(1/p, n)); while(x<1, x*=10); floor(x); \\ Michel Marcus, Jun 29 2025

Formula

a(n) = most significant (nonzero) digit of P(n), where P(n) := Sum_{p prime} 1/p^n.

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.

For all n > 9, a(n) = most significant (nonzero) digit of 5^n (conjectured).