Prime Zeta Function

The prime zeta function is defined for by

where the sum is taken over primes. It is a generalization of the Riemann zeta function

where the series also requires and the sum is over all positive integers. In other words, the prime zeta function is the Dirichlet generating function of the characteristic function of the primes . is illustrated above on the positive real axis, where the imaginary part is indicated in yellow and the real part in red. (The sign difference in the imaginary part compared to the plot appearing in Fröberg is presumably a result of the use of a different convention for .)

Various terms and notations are used for this function. The term "prime zeta function" and notation were used by Fröberg (1968), whereas Cohen (2000) uses the notation .

The series converges absolutely for , where , can be analytically continued into the strip away from its singularities, with consistent choices of logarithm branches (Fröberg 1968), but not beyond the line (Landau and Walfisz 1920, Fröberg 1968) due to the clustering of singular points along the imaginary axis arising from the nontrivial zeros of the Riemann zeta function on the critical line .

As illustrated in the left figure above (where the real part is indicated in red and the imaginary part in yellow), the function has singular points along the real axis for where runs through all squarefree positive integers. For close to 1, has the expansion

where and

(OEIS A143524), where is the Möbius function and is the Riemann zeta function (Fröberg 1968).

The prime zeta function is plotted above for and (Fröberg 1968).

The prime zeta function is illustrated above in the complex plane.

The prime zeta function can be expressed in terms of the Riemann zeta function for by

Inverting then gives

(Glaisher 1891, Fröberg 1968, Cohen 2000).

The prime zeta function is implemented in the Wolfram Language as PrimeZetaP[s].

The Dirichlet generating function of the composite numbers is given for by

The series for , the analog of the harmonic series, diverges, whereas that for converges absolutely for . The convergent sum over integer arguments gives the Mertens constant

(OEIS A077761).

Artin's constant is connected with by

where is a Lucas number (Ribenboim 1998, Gourdon and Sebah).

The values of for the first few integers starting with two are given in the following table. Merrifield (1881) computed for up to 35 to 15 digits, and Liénard (1948) computed up to to 50 digits (Ribenboim 1996). Gourdon and Sebah give values to 60 digits for .

According to Fröberg (1968), very little was known about the roots of . The plots above show the positions of zeros (left figure) and zero contours of the real part (red) and imaginary part (blue) in a portion of the complex plane, with roots indicated as black dots (right figure).