The Primezeta Function
Z, PZ, CZ
Im(s)=π/2 for s<700 s=zero of PZ
Im(s)=π/2 for s<700 s=zero of PZ
The relationship between Z and PZ can be established using:
PZ(s) = sum {k=1,∞} µ(k) * log(Z(k*s))/k
In the attached chart, it shows how close are Z and exp(PZ) for s>2.
In 2017, I introduced an approximation CZeta(s) for Z(s) in the form:
CZ(s) = 1 / ( 1 -π^-s - 2^-s)
the chart shows how CZ approaches Z and e^PZ for values s>2.
Im(s)=π/2 for s<700 s=zero of PZ
Im(s)=π/2 for s<700 s=zero of PZ
Im(s)=π/2 for s<700 s=zero of PZ
The Im(s) for the zeros of PZ less than 1000 is π/2
Ln(z*) for the NT zeros of Zeta
Im(s)=π/2 for s<700 s=zero of PZ
Ln(z*) for the NT zeros of Zeta
The trend line for the logarithms of zeta at the non-trivial zeros.
Zeros of PZ
Zeros of PZ | Re(s)<3 Im(s)<800
Ln(z*) for the NT zeros of Zeta
Full chart of the first 1000 zeros of PZ
Zeros of PZ | Re(s)<3
Zeros of PZ | Re(s)<3 Im(s)<800
Zeros of PZ | Re(s)<3 Im(s)<800
Zeros of PZ with Re(s)<3
Zeros of PZ | Re(s)<3 Im(s)<800
Zeros of PZ | Re(s)<3 Im(s)<800
Zeros of PZ | Re(s)<3 Im(s)<800
Zeros of PZ with Re(s)<3 and Im(s)<800.
Zeros of PrimeZeta
Zeros of PrimeZeta in Log Scales
Zeros of PrimeZeta in Log Scales
Zeros of PrimeZeta in Log Scales
When representing Im(s) vs. Re(s) for s=zero of PZ, four clusters emerge.
Log(zeros) of PZ
Zeros of PrimeZeta in Log Scales
Zeros of PrimeZeta in Log Scales
Im(log(s)) vs. Re(log(s))