A385495 - OEIS

A385495

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

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, 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

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OFFSET

2,1

COMMENTS

The sequence starts at n = 2 since the zeta function sum diverges for any integer n < 2.

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).

REFERENCES

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

LINKS

Table of n, a(n) for n=2..88.

Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.

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?.

Eric Weisstein's World of Mathematics, Riemann Zeta Function.

Wikipedia, Riemann Zeta function.

EXAMPLE

For n = 4, a(4) = 8 since the most significant digit of zeta(4)-1 = 0.0823232... is 8 (see A013662).

MATHEMATICA

Table[First@Select[First@RealDigits[N[Zeta[n] - 1, 100]], # != 0 &], {n, 2, 100}]