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As other answers have mentioned, the most "natural" value for the harmonic series seems to be the Euler-Mascheroni constant $\gamma$. The article Euler constant as a renormalized value of Riemann zeta function at its pole by Andrei Vieru says

We believe that Euler constant is not just the "renormalized" value of the Riemann zeta function in 1. In a sense that we shall clarify it is in fact the normal and natural value of zeta of 1. In this paper we first propose a limit definition of a function whose values coincide everywhere with those of the Riemann zeta function, save in 1, where our limit definition yields the Euler constant. Since in the literature one can find more than one way to regularize the value of the zeta function at s=1, we give asymptotic expansions where, by dint of some extended analogies, Euler constant appears to be the true "renormalized" value. As a striking example of such analogies, we propose an expansion of the logarithm function based on Euler constant and on all values of the zeta function at odd positive integers, in which all these presumably irrational numbers are accompanied by Harmonic numbers of corresponding orders. The other aim of this paper is to show how sequences of rationals, often the same, arise in computations related to Dirichlet L-functions. Here, a connection with the Liouville lambda function appears to have been found. Thus we raise the question about the possible usefulness of an extension of the Liouville lambda function to rationals.

One example given in the article (on page 4, equation 11) is

$$\ln \Gamma(x) = -\ln x - \gamma x + \sum_{k \geq 2} \frac{\zeta(k)}{k}(-x)^k$$

If we let $\zeta(1) = \gamma$, this can be simplified to

$$\sum_{k \geq 1} \frac{\zeta(k)}{k} x^k = \ln (-x)!$$

which is, I think, supremely elegant. More generally, we seem to have

$$\sum_{k \geq 1} \zeta(k) k^n x^k = \sum_{k=0}^n \left\{{n+1 \atop k+1}\right\} (-x)^{k+1} \psi^{(k)}(1-x)$$

where $\psi^{(k)}$ is the polygamma function.

As other answers have mentioned, the most "natural" value for the harmonic series seems to be the Euler-Mascheroni constant $\gamma$. The article Euler constant as a renormalized value of Riemann zeta function at its pole by Andrei Vieru says

We believe that Euler constant is not just the "renormalized" value of the Riemann zeta function in 1. In a sense that we shall clarify it is in fact the normal and natural value of zeta of 1. In this paper we first propose a limit definition of a function whose values coincide everywhere with those of the Riemann zeta function, save in 1, where our limit definition yields the Euler constant. Since in the literature one can find more than one way to regularize the value of the zeta function at s=1, we give asymptotic expansions where, by dint of some extended analogies, Euler constant appears to be the true "renormalized" value. As a striking example of such analogies, we propose an expansion of the logarithm function based on Euler constant and on all values of the zeta function at odd positive integers, in which all these presumably irrational numbers are accompanied by Harmonic numbers of corresponding orders. The other aim of this paper is to show how sequences of rationals, often the same, arise in computations related to Dirichlet L-functions. Here, a connection with the Liouville lambda function appears to have been found. Thus we raise the question about the possible usefulness of an extension of the Liouville lambda function to rationals.

One example given in the article (on page 4, equation 11) is

$$\ln \Gamma(x) = -\ln x - \gamma x + \sum_{k \geq 2} \frac{\zeta(k)}{k}(-x)^k$$

where $\Gamma$ is the gamma function and $\zeta$ is the zeta function. If we let $\zeta(1) = \gamma$, this can be simplified to

$$\sum_{k \geq 1} \frac{\zeta(k)}{k} x^k = \ln (-x)!$$

which is, I think, supremely elegant. More generally, we seem to have

$$\sum_{k \geq 1} \zeta(k) k^n x^k = \sum_{k=0}^n \left\{{n+1 \atop k+1}\right\} (-x)^{k+1} \psi^{(k)}(1-x)$$

where $\psi^{(k)}$ is the polygamma function and the brackets indicate Stirling numbers of the second kind.

As other answers have mentioned, the most "natural" value for the harmonic series seems to be the Euler-Mascheroni constant $\gamma$. The article Euler constant as a renormalized value of Riemann zeta function at its pole by Andrei Vieru says

We believe that Euler constant is not just the "renormalized" value of the Riemann zeta function in 1. In a sense that we shall clarify it is in fact the normal and natural value of zeta of 1. In this paper we first propose a limit definition of a function whose values coincide everywhere with those of the Riemann zeta function, save in 1, where our limit definition yields the Euler constant. Since in the literature one can find more than one way to regularize the value of the zeta function at s=1, we give asymptotic expansions where, by dint of some extended analogies, Euler constant appears to be the true "renormalized" value. As a striking example of such analogies, we propose an expansion of the logarithm function based on Euler constant and on all values of the zeta function at odd positive integers, in which all these presumably irrational numbers are accompanied by Harmonic numbers of corresponding orders. The other aim of this paper is to show how sequences of rationals, often the same, arise in computations related to Dirichlet L-functions. Here, a connection with the Liouville lambda function appears to have been found. Thus we raise the question about the possible usefulness of an extension of the Liouville lambda function to rationals.

One example given in the article (on page 4, equation 11) is

$$\ln \Gamma(x) = -\ln x - \gamma x + \sum_{k \geq 2} \frac{\zeta(k)}{k}(-x)^k$$

If we let $\zeta(1) = \gamma$, this can be simplified to

$$\sum_{k \geq 1} \frac{\zeta(k)}{k} x^k = \ln (-x)!$$

which is, I think, supremely elegant.

As other answers have mentioned, the most "natural" value for the harmonic series seems to be the Euler-Mascheroni constant $\gamma$. The article Euler constant as a renormalized value of Riemann zeta function at its pole by Andrei Vieru says

We believe that Euler constant is not just the "renormalized" value of the Riemann zeta function in 1. In a sense that we shall clarify it is in fact the normal and natural value of zeta of 1. In this paper we first propose a limit definition of a function whose values coincide everywhere with those of the Riemann zeta function, save in 1, where our limit definition yields the Euler constant. Since in the literature one can find more than one way to regularize the value of the zeta function at s=1, we give asymptotic expansions where, by dint of some extended analogies, Euler constant appears to be the true "renormalized" value. As a striking example of such analogies, we propose an expansion of the logarithm function based on Euler constant and on all values of the zeta function at odd positive integers, in which all these presumably irrational numbers are accompanied by Harmonic numbers of corresponding orders. The other aim of this paper is to show how sequences of rationals, often the same, arise in computations related to Dirichlet L-functions. Here, a connection with the Liouville lambda function appears to have been found. Thus we raise the question about the possible usefulness of an extension of the Liouville lambda function to rationals.

One example given in the article (on page 4, equation 11) is

$$\ln \Gamma(x) = -\ln x - \gamma x + \sum_{k \geq 2} \frac{\zeta(k)}{k}(-x)^k$$

If we let $\zeta(1) = \gamma$, this can be simplified to

$$\sum_{k \geq 1} \frac{\zeta(k)}{k} x^k = \ln (-x)!$$

which is, I think, supremely elegant. More generally, we seem to have

$$\sum_{k \geq 1} \zeta(k) k^n x^k = \sum_{k=0}^n \left\{{n+1 \atop k+1}\right\} (-x)^{k+1} \psi^{(k)}(1-x)$$

where $\psi^{(k)}$ is the polygamma function.

As other answers have mentioned, the most "natural" value for the harmonic series seems to be the Euler-Mascheroni constant $\gamma$. The article Euler constant as a renormalized value of Riemann zeta function at its pole by Andrei Vieru says

We believe that Euler constant is not just the "renormalized" value of the Riemann zeta function in 1. In a sense that we shall clarify it is in fact the normal and natural value of zeta of 1. In this paper we first propose a limit definition of a function whose values coincide everywhere with those of the Riemann zeta function, save in 1, where our limit definition yields the Euler constant. Since in the literature one can find more than one way to regularize the value of the zeta function at s=1, we give asymptotic expansions where, by dint of some extended analogies, Euler constant appears to be the true "renormalized" value. As a striking example of such analogies, we propose an expansion of the logarithm function based on Euler constant and on all values of the zeta function at odd positive integers, in which all these presumably irrational numbers are accompanied by Harmonic numbers of corresponding orders. The other aim of this paper is to show how sequences of rationals, often the same, arise in computations related to Dirichlet L-functions. Here, a connection with the Liouville lambda function appears to have been found. Thus we raise the question about the possible usefulness of an extension of the Liouville lambda function to rationals.

One example given in the article (on page 4, equation 11) is

$$\ln \Gamma(x) = -\ln x - \gamma x + \sum_{k \geq 2} \frac{\zeta(k)}{k}(-x)^k$$

If we let $\zeta(1) = \gamma$, this can be simplified to

$$\sum_{k \geq 1} \frac{\zeta(k)}{k} x^k = \ln (-x)!$$

which is, I think, supremely elegant. The choice $\zeta(1) = \gamma$ also simplifies the following formula:

$$\sum_{k \geq 1} \frac{\zeta(k)-1}{k} x^k = \ln (1-x)!$$

As other answers have mentioned, the most "natural" value for the harmonic series seems to be the Euler-Mascheroni constant $\gamma$. The article Euler constant as a renormalized value of Riemann zeta function at its pole by Andrei Vieru says

We believe that Euler constant is not just the "renormalized" value of the Riemann zeta function in 1. In a sense that we shall clarify it is in fact the normal and natural value of zeta of 1. In this paper we first propose a limit definition of a function whose values coincide everywhere with those of the Riemann zeta function, save in 1, where our limit definition yields the Euler constant. Since in the literature one can find more than one way to regularize the value of the zeta function at s=1, we give asymptotic expansions where, by dint of some extended analogies, Euler constant appears to be the true "renormalized" value. As a striking example of such analogies, we propose an expansion of the logarithm function based on Euler constant and on all values of the zeta function at odd positive integers, in which all these presumably irrational numbers are accompanied by Harmonic numbers of corresponding orders. The other aim of this paper is to show how sequences of rationals, often the same, arise in computations related to Dirichlet L-functions. Here, a connection with the Liouville lambda function appears to have been found. Thus we raise the question about the possible usefulness of an extension of the Liouville lambda function to rationals.

One example given in the article (on page 4, equation 11) is

$$\ln \Gamma(x) = -\ln x - \gamma x + \sum_{k \geq 2} \frac{\zeta(k)}{k}(-x)^k$$

If we let $\zeta(1) = \gamma$, this can be simplified to

$$\sum_{k \geq 1} \frac{\zeta(k)}{k} x^k = \ln (-x)!$$

which is, I think, supremely elegant.