In[]:=
Prime[6]
Out[]=
13

Aleternatinglyaddallprimesuptothe
x'th
10
primethensubracting(qb=atermofthe ThueConstantandtheMRBconstant).​​CMRB=
∞
∑
i=1
i
(-1)
(
1/i
i
-1)​​p(x)=
x
10
∑
i=1
i+1
(-1)
p
i
;​​q=
p
2
p
2
+1
∏
i=
p
1
p
i
;​​b=p-
p
1
p
9
q(-
p
1
+(
p
9
-
p
1
)ch)
-CMRB;​​qb-p(x)gives​​
​where CMRB is the MRB constant,​
p
i
is the i’th prime and ch is the
ThueConstantinbase10
.

Let ch be the Thue-Morse Constant in base 10.
Letchbethe
Thue-MorseConstant
inbase10.
In[]:=
tm=Nest[Flatten[#/.{0{1,1,1},1{1,1,0}}]&,{0},10];
In[]:=
ch=FromDigits[RealDigits[FromDigits[{tm,0},2]],10];
Let p be the partial sums of the alternating Harmonic series over primes to the 10^13th prime:
In[]:=
p=TableNSum
(-1)^(i+1)
Prime[Floor[i]]
,{i,1,
n
10
},Method"AlternatingSigns",WorkingPrecision30,{n,2,13};
Let CMRB be the MRB constant:
∞
∑
i=1
i
(-1)
(
1/i
i
-1)
In[]:=
CMRB=NSum[(-1)^i
1/(i)
(i)
-1,{i,1,Infinity},Method"AlternatingSigns",WorkingPrecision30]
Out[]=
0.18785964246206712024857897184
In[]:=
q=
p
2
p
2
+1
∏
i=
p
1
p
⌊i⌋
;
In[]:=
b=p-
p
1
p
9
q(-
p
1
+(
p
9
-
p
1
)ch)
-CMRB
Out[]=
{0.07209673976214940650175485837,0.07295277329599462284881235013,0.07301110018003829179202736209,0.07301548830561549825187221943,0.07301584073079151856660077498,0.07301587023167460087096914492,0.07301587277303004072035826393,0.07301587299643149339488140551,0.07301587301637626461625166922,0.07301587301817851017013049613,0.07301587301834295310015652105,0.07301587301835807760869217212}
In[]:=
TableForm{Range[12],Abs[N[qb-
p
9
,3]]},TableHeadings"x=I'th prime","absq b-
x
10
∑
i=1
i+1
(-1)
p
i
​",None
Out[]//TableForm=
x=I'th prime
1
2
3
4
5
6
7
8
9
10
11
12
absq b-
x
10
∑
i=1
i+1
(-1)
p
i
​
0.290
0.0199
0.00150
0.000121
0.0000102
8.77×
-7
10
7.65×
-8
10
6.12×
-9
10
1.59×
-10
10
7.26×
-10
10
7.78×
-10
10
7.83×
-10
10