Very often, my good friend Aurel comes to me with some mathematical facts more or less funny, but always interesting. This time he let me know about something which is called Chebyshev Bias.
Chebyshev noted in 1853, that if you take modulo 4 of the prime numbers, the which gives 1 as rest are less than the ones which gives rest 3. We denote by \(\pi(x,4,1)\) primes \(p \leq x \), congruent with 1 modulo 4 and with \(\pi(x,4,3)\) primes \(p \leq x\), congruent with 3 modulo 4.
It is a known fact that \(\pi(x,a,b) \sim \frac{1}{\varphi(q)}\frac{x}{ln(x)}\). That means that this number does not depend on the value of \(a\). As a consequence, one would expect that \(\pi(x,4,1) \sim \pi(x,4,3)\). Which is true when \(x \to \infty\).
If we draw the ration between \(\pi(x,4,3)\) and \(\pi(x,4,1)\) we can start to see what Chebyshev noted. The ratio gets closer to 1, but it seems to be almost always a little bit upper than that.
But we can get a better picture if we plot the difference between those values, not the ratio. We denote by \(delta(x) = \pi(x,4,3) - \pi(x,4,1) \). If we plot delta values for \( x \leq 5000\) we have the plot below.
It is obvious that the value of delta function is always greater or equal than 0 with one exception. This exception was determined also by Chebyshev, and the value is 2946. But this is the only exception? We can see something if we plot delta values for \(x\leq 60000\).
It is obvious that there are some points where delta function is negative, and those points are around 50380. This time there are many points not just one.
If I plot the delta for all the prime number values that I have (in a file computed with a naive sieve), we see that the next time when the delta function has negative values is somewhere around 48 millions, which is preety high.
It is sure that we can't draw any conclusion from those plots. The delta function looks like one which is able to provide unexpected behavior. Howevere it seems safe to assume that most of the time the delta function has a positive value. For me, at least, this looks like an interesting and unexpected fact.
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