<p>Engineer.</p>
<p><span class="math-container">$T(n,k)$</span> = <a href="https://oeis.org/A191898" rel="nofollow noreferrer">https://oeis.org/A191898</a> <span class="math-container">$(n,k)$</span></p>
<p><span class="math-container">$$\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(s-1)}}$$</span>
<span class="math-container">$$\sum\limits_{k=1}^{\infty}\sum\limits_{n=1}^{\infty} \frac{T(n,k)}{n^c \cdot k^s} = \sum\limits_{n=1}^{\infty} \frac{\lim\limits_{z \rightarrow s} \zeta(z)\sum\limits_{d|n} \frac{\mu(d)}{d^{(z-1)}}}{n^c} = \frac{\zeta(s) \zeta(c)}{\zeta(s + c - 1)}$$</span></p>
<p><span class="math-container">$$-\frac{\zeta '(s)}{\zeta (s)}=\lim_{c\to 1} \, \left(\frac{\zeta (s) \zeta (c)}{\zeta (s+c-1)}-\zeta (c)\right)$$</span></p>
<p><span class="math-container">$$\mu(n) =  \underbrace{\underset{1 = n} 1 - \underset{a = n}{\sum_{a \geq 2}} 1 + \underset{ab = n}{\sum_{a \geq 2} \sum_{b \geq 2}} 1 - \underset{abc = n}{\sum_{a \geq 2} \sum_{b \geq 2} \sum_{c \geq 2}} 1 + \underset{abcd = n}{\sum_{a \geq 2} \sum_{b \geq 2} \sum_{c \geq 2} \sum_{d \geq 2}} 1 - \cdots}_{\text{#alternating sums}&gt;\frac{\log(n)}{\log(2)}}$$</span></p>
<p><span class="math-container">$$1/a^{b+i c}=1/a^b (\cos (c \log (1/a))+i \sin (c \log (1/a)))$$</span></p>
<pre><code>1/a^(b + I*c) = 1/a^b*(Cos[c*Log[1/a]] + I*Sin[c*Log[1/a]])
</code></pre>
<hr />
<p><span class="math-container">$$f(n,s)=\frac{(s+1)^{n-1}+s-1}{s}$$</span></p>
<pre><code>N[Table[2*Pi*Exp[1]*Exp[ProductLog[(n - 11/8)/Exp[1]]], {n, 1, 12}]]

Limit[(Zeta[s]*Zeta[s]/Zeta[2 s - 1] + Zeta[s]/Zeta[s - 1]), s -&gt; 1]

Plot[RiemannSiegelTheta[t]/Pi + 
  Im[Log[Zeta[1/2 + I*t]] + I*Pi]/Pi, {t, 0, 60}, ImageSize -&gt; Large]

Table[Limit[
  Zeta[s] Total[1/Divisors[n]^(s - 1)*MoebiusMu[Divisors[n]]], 
  s -&gt; 1], {n, 1, 32}]

x = N[Exp[-ZetaZero[1]/10], 100]
Sum[(-1)^k*x^(Log[k]*10), {k, 1, Infinity}]
</code></pre>
<p><span class="math-container">$$\sum _{k=1}^{\infty } (-1)^{k+1} x^{c \log (k)}=0?$$</span>
Keith Conrad: numbersoncircle.pdf</p>
<p>=IF(OR(ROW()=1; COLUMN()=1);0; IF(ROW()&gt;=COLUMN();EXP(-(1-11/8/(COLUMN()-1))/EXP(1)*SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1; COLUMN(); 4)&amp;&quot;:&quot;&amp;ADDRESS(ROW()-1; COLUMN(); 4); 4)));0))</p>
<p>divided with: /2/PI()/EXP(1)</p>
<p>von Mangoldt function:</p>
<p>=IF(OR(ROW()=1, COLUMN()=1), 1, IF(ROW()&gt;=COLUMN(),-SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1,COLUMN(), 4)&amp;&quot;:&quot;&amp;ADDRESS(ROW()-1, COLUMN(), 4), 4)),-SUM(INDIRECT(ADDRESS(COLUMN()-ROW()+1,ROW(), 4)&amp;&quot;:&quot;&amp;ADDRESS(COLUMN()-1, ROW(), 4), 4))))</p>
<p><a href="http://pastebin.com/u/MatsGranvik" rel="nofollow noreferrer">http://pastebin.com/u/MatsGranvik</a></p>
<p>Divisibility:</p>
<p>=IF(OR(COLUMN()=1), 1, IF(ROW()&gt;=COLUMN(),SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1,COLUMN()-1, 4)&amp;&quot;:&quot;&amp;ADDRESS(ROW()-1, COLUMN()-1, 4), 4))-SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1,COLUMN(), 4)&amp;&quot;:&quot;&amp;ADDRESS(ROW()-1, COLUMN(), 4), 4)),0))</p>
<p><span class="math-container">$$\mu(n) =  \underbrace{\underset{1 = n} 1 - \underset{a = n}{\sum_{2 \leq a \leq \frac{n}{2^0} }} 1 + \underset{ab = n}{\sum_{2 \leq a \leq \frac{n}{2^1}} \sum_{2 \leq b \leq \frac{n}{2^1}}} 1 - \underset{abc = n}{\sum_{2 \leq a \leq \frac{n}{2^2}} \sum_{2 \leq b \leq \frac{n}{2^2}} \sum_{2 \leq c \leq \frac{n}{2^2}}} 1 + \underset{abcd = n}{\sum_{2 \leq a \leq \frac{n}{2^3}} \sum_{2 \leq b \leq \frac{n}{2^3}} \sum_{2 \leq c \leq \frac{n}{2^3}} \sum_{2 \leq d \leq \frac{n}{2^3}}} 1 - \cdots}_{\text{#alternating sums}&gt;\frac{\log(n)}{\log(2)}}$$</span></p>

Mats Granvik