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DNA Primes Math Blog

DNA Primes Math BlogDNA Primes Math BlogDNA Primes Math Blog
  • Home
  • My Blog
  • OEIS
  • Papers
  • Prime Numbers
  • Riemann Zeta
  • Infinite Sums/Products
  • PrimeZeta
  • Complex Functions

The Primezeta Function

Z, PZ, CZ

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

The Im(s) for the zeros of PZ less than 1000 is π/2

Ln(z*) for the NT zeros of Zeta

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))

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