I have found the following formulæ incidentally in the course of other investigations. None of them seem to be of particular importance, nor does their proof involve the use of any new ideas, but some of them are so curious that they seem to be worth printing. I denote by $d(x)$ the number of divisors of $x$, if $x$ is an integer, and zero otherwise, and by $\zeta(s)$ the Riemann Zeta-function.

where $$\eta (s)=1^{-s}-3^{-s}+5^{-s}-7^{-s}+\cdots~.$$ where $$ A = \frac{1}{\pi^2}, \: B = \frac{12 \gamma - 3}{\pi^2}- \frac{36}{\pi^4} \zeta' (2),$$ $\gamma$ is Euler's constant, $C, D$ more complicated constants, and $\epsilon$ any positive number. where $\phi(s)$ is absolutely convergent for $R(s) > \frac{1}{2}$, and in particular where $$A_1 = \frac{1}{\sqrt{\pi}} \prod_p \left\{\sqrt{(p^2 - p)} \log \left(\frac{p}{p-1}\right)\right\}$$ and $A_2, A_3, \ldots A_r$ are more complicated constants.

More generally

if $2^s$ is an integer, and if $2^s$ is not an integer, the $A$'s being constants. where $$C = \gamma + \sum^\infty_2 \left\{ \log_2 \left(1 + \frac{1}{\nu}\right) - \frac{1}{\nu} \right\} (2^{-\nu} + 3^{-\nu} + 5^{-\nu} + \ldots ).$$ Here $2,3,5, \ldots$ are the primes and \begin{eqnarray*} \frac{\phi(n)}{n} = \frac{\gamma -1}{\log n} + \frac{1!}{(\log n)^2} (\gamma + \gamma_1 -1 ) + \frac{2!}{(\log n)^3} (\gamma + \gamma_1 + \gamma_2 - 1) + \ldots \cr + \frac{(r-1)!}{(\log n)^r} (\gamma + \gamma_1 + \gamma_2 + \ldots + \gamma_{r-1} - 1) + O \left\{\frac{1}{(\log n)^{r+1}} \right\}, \end{eqnarray*} where $$ \zeta (1+s) = \frac{1}{s} + \gamma - \gamma_1 s + \gamma_2 s^2 - \gamma_3 s^3 + \ldots $$ or $$ r ! \gamma_r = \lim_{\nu \to \infty} \left\{ (\log 1)^r + \frac{1}{2} (\log 2)^r + \ldots + \frac{1}{\nu} (\log \nu)^r - \frac{1}{r+1} (\log \nu)^{r+1} \right\}. $$ where $\delta$ is a common factor of $u$ and $v$, and $$ \frac{1}{\zeta (s)} = \sum^\infty_1 \frac{\mu (n)}{n^s}.$$ $$ {\rm (G)}\:\:\:\:\:\:\:\:\:\:\:{\rm If}\:\:\: D_v(n) = d(v) + d(2v) + \ldots + d(nv), $$ we have where $\delta$ is a divisor of $v$, and where $$ \sum^\infty_1 \frac{\alpha(\nu)}{\nu^s} = \frac{\zeta^2(s)}{\zeta(1+s)}, \sum^\infty_1 \frac{\beta(\nu)}{\nu^s} = -\frac{\zeta^2 (s) \zeta' (1+s)}{\zeta^2 (1+s)}, $$ and $$ \Delta_v (n) = O (n^{\frac{1}{3}} \log n)~\href{#p17_en4}{^4}$$ where $$ \sum^\infty_1 \frac{\alpha_c (\nu)}{\nu^s} = \frac{\zeta(s) \sigma_{-s} (|c|)}{\zeta(1+s)},$$ $$\sum^\infty_1 \frac{\beta_c (\nu)}{\nu^s} = \frac{\zeta(s) \sigma_{-s} (|c|)}{\zeta(1+s)} \left\{\frac{\zeta' (s)}{\zeta(s)} + \frac{\zeta' (1+s)}{\zeta (1+s)} + \frac{\sigma_{-s}~'(|c|)}{\sigma_{-s} (|c|)}\right\}, $$ $\sigma_s (n)$ being the sum of the $s$th powers of the divisors of $n$ and $\sigma_s'(n)$ the derivative of $\sigma_s(n)$ with respect to $s$, and $$ \Delta_{v,c} (n) = O (n^{\frac{1}{3}} \log n).\href{#p17_en15}{^5} $$ (I) The formulæ (1) and (2) are special cases of It is possible to find an approximate formula for the general sum The general formula is complicated, The most interesting cases are $a=0, b=0$, when the formula is (3); $a = 0, b=1,$ when it is where $$ c = \gamma - \frac{1}{4} + \frac{\zeta'(2)}{\zeta(2)} - \frac{\zeta'(3)}{\zeta(3)}, $$ and the order of $E(n)$ is the same as that of $\Delta_1(n)$; and $a=1, b=1$, when it is where $$E(n) = O \{n^2 (\log n)^2\}, \: E(n) \neq o (n^2 \log n).$$ $\mbox{(J)}~~ \mbox{If } s>0$, then where $$ 1 > \theta > (1-2^{-s}) (1 - 3^{-s}) (1 - 5^{-s}) \ldots (1- \varpi^{-s}), $$ $\varpi$ is the greatest prime not exceeding $n$, and $$ c = \prod_p \left\{ \left( \frac{p^{2s} -1}{p^{2s} - p^s}\right)^{1/p} \left(\frac{p^{3s}-1}{p^{3s} - p^s} \right)^{1/p^2} \left(\frac{p^{4s}-1}{p^{4s}-p^s}\right)^{1/p^3} \cdots \right\}. $$ $$\mbox{(K)}\:\:\:\:\:\:\:\mbox{If}\:\:\:\:\:\:\:\left(\frac{1}{2} + q + q^4 + q^9 + q^{16} + \cdots\right)^2 = \frac{1}{4} + \sum^\infty_1 r (n) q^n, $$ so that $$ \zeta (s) \eta(s) = \sum^\infty_1 r (n) n^{-s}, $$ then where $$ C = 4 \gamma - 1 + \frac{1}{3} \log 2 - \log \pi + 4 \log \Gamma (\frac{3}{4}) - \frac{12}{\pi^2} \zeta' (2). $$ These formulæ are analogous to (1) and (3).