13 Comparison of Fourier Series Representations of Prime Counting Functions

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On page 1 I mentioned I’ve defined three methods for derivation of Fourier series, and this web site primarily illustrates the first method. The comparison below is with respect to Fourier series derived for each of the prime counting functions via the first method.


I think the simple prime-power counting function $K[x]$ is more fundamental in a couple of ways than the base prime counting function $\pi[x]$ and Riemann’s prime-power counting function $J[x]$ which I attempt to illustrate in the two items below.


  • First, the coefficients for converting between the simple prime-power counting function $K[x]$ and the base prime counting function $\pi[x]$ are simpler than the coefficients for converting between Riemann’s prime-power counting function $J[x]$ and the base prime counting function $\pi[x]$ (i.e. the $\frac{1}{n}$ term is removed). On the other hand, the coefficients which I’ve derived for converting between the simple prime-power counting function $K[x]$ and Riemann’s prime-power counting function $J[x]$ illustrated on page 3-3 are more complex, but personally I find mathematical beauty in these coefficients despite their increased complexity, and I’d argue the complexity of these coefficients is due to the complexity of $J[x]$ verus the complexity of $K[x]$.

  • Second, the simple prime-power counting function $K[x]$ has the simplest Fourier series, and the Fourier series for the base prime counting function $\pi[x]$ and Riemann’s prime-power counting function $J[x]$ were essentially derived from the Fourier series for the simple prime-power counting function $K[x]$. The order of complexity of these three functions from simplest to most complicated is $K[x]<\pi[x]<J[x]$.

With respect to the remaining three prime counting functions, the second Chebyshev function $\psi[x]$ seems to be the most fundamental since it has the simplest Fourier series, and the Fourier series for the first Chebyshev function $\vartheta[x]$ and third “Chebyshev-like” function $M[x]$ were essentially derived from the Fourier series for the second Chebyshev function $\psi[x]$. The order of complexity of these three functions from simplest to most complicated is $\psi[x]<M[x]<\vartheta[x]$. It’s interesting the second Chebyshev function $\psi[x]$ is the simplest of these three functions, whereas the related Riemann prime-power counting function $J[x]$ (recall $\psi'[x]=\log[x]\ J'[x]$) is the most complicated of the other three prime counting functions $\pi[x]$, $J[x]$, and $K[x]$.


The Fourier series for the prime counting functions are compared further in the items below.


  • The Fourier series for the base prime counting function $\pi[x]$ is the most ideal because of a couple of factors including the uniform unit step size at each prime and the slower growth rate of this function relative to Riemann’s prime-power counting function $J[x]$ and the simple prime-power counting function $K[x]$. Evaluation parameters can be selected such that $\pi[0]=0$ (at least for small ranges of $x$ values), in which case $\pi[x]$ evaluates to an odd function (i.e. $\pi[-x]=-\pi[x]$). Selection of evaluation parameters such that $\pi[0]=0$ also minimizes the offset of the evaluation of $\pi'[x]$ at primes from the peaks of their associated primary lobes. The question of whether evaluation parameters can always be selected such that $\pi[0]=0$ for any arbitrarily large range of $x$ values is of theoretical interest, and further investigation is needed to answer this question.

  • The Fourier series for Riemann’s prime-power counting function $J[x]$ is less than ideal for a couple of reasons including the variable $\frac{1}{n}$ step size at prime powers of the form $x=p^n$, and the faster growth of this function compared with the base prime counting function $\pi[x]$. The first factor makes it impossible to select evaluation parameters such that $J[0]=0$, and consequently $J[x]$ can never be evaluated exactly to an odd function. The second factor makes it more difficult to select evaluation parameters to minimize the offset of the evaluation of $J'[x]$ at prime-powers from their associated primary lobes when evaluating larger ranges of $x$ values.

  • The Fourier series for the simple prime-power counting function $K[x]$ is less than ideal because the faster growth of this function compared with the base prime counting function $\pi[x]$. This makes it possible to select evaluation parameters such that $K[0]=0$ only for very small ranges of $x$ values, and consequently the Fourier series $K[x]$ can only be evaluated exactly to an odd function for very small ranges of $x$ values. This also makes it more difficult to select evaluation parameters to minimize the offset of the evaluation of $K[x]$ at prime-powers from their associated primary lobes when evaluating larger ranges of $x$ values.

  • The Fourier series for the first Chebyshev function $\vartheta[x]$, second Chebyshev function $\psi[x]$, and third “Chebyshev-like” function $M[x]$ are less than ideal because of the $\log$ nature of their step sizes at primes (and in some cases prime-powers). This makes it impossible to select evaluation parameters such that the evaluations of $\vartheta[x]$, $\psi[x]$, and $M[x]$ at $x=0$ are exactly equal to zero, and consequently $\vartheta[x]$, $\psi[x]$, and $M[x]$ can never be evaluated exactly to odd functions.