If the Riemann hypothesis is wrong…

As you know, in wartime, the value of the cosine can reach three. Fortunately, this does not apply to the simplicity of numbers – no matter how you beat your forehead against the wall, the number 17 is prime and is not divisible by anything except itself and 1.

Or not? What if we rudely crowbar through the holy of holies of mathematics and move the non-trivial zeros of the zeta function? Will the prime numbers move from their places? Pictures and videos are waiting for you, and very few formulas.

I will skip the popular exposition of the Riemann Hypothesis – it has been covered an unthinkable number of times. Let’s go straight to the end: knowing the non-trivial zeros of the zeta function, we can ‘reverse’ write a function that represents pi(x) – ‘prime counting function‘ is a function that increments by one on every prime number:

There are also small steps, they must be ignored, they correspond, in particular, to the powers of prime numbers. If we ‘catch’ large steps, then we get magic:

After an inconceivably large number of floating point operations, logarithms and cosines, we get prime numbers … Something that can be counted on the fingers, but the magic is that this time they come from the side, where, in general, with number theory nothing is connected.

I took formulas for calculation here and there is also an interesting animation. So, the main ‘move’ of the pi(x) function is given by:

This part is corrected by the term responsible for trivial and nontrivial zeros (lower formula):

For some reason, with the second expression, I got very large values ​​\u200b\u200b(alas, I didn’t understand why) and I used an expression that should be equivalent (only named C and not T), for C, you need to take the sum over all non-trivial roots:

I downloaded 100K roots and ran the program. Now it’s time for the crowbar. Let’s take the first root and shift it ‘up’ (that is, increasing the imaginary part, and decreasing for the conjugate root):

Now let’s take and move ALL the roots like this:

Note that the steps are ‘deteriorated’, but remain in the same places. You can shift prime numbers only if you multiply the imaginary part of non-trivial roots by a small constant:

Finally, let the Riemann hypothesis be wrong. Let us shift the real part of all nontrivial roots from their place, which is known to be 1/2:

Alas. There will be no beautiful animation. Nothing has changed, except for the third – fourth digit after the decimal point. However, this is not surprising: if you look at the formulas, then the only place where the real part of the root is important is the last formula, arg(Pk)but arg (phase) is already almost vertical – even for the first root (0.5+14.13i) the imaginary part is much larger than the real part. Please note that the picture above is not to scale. (However, maybe the last formula was derived under the assumption that the real part is 0.5? Let more advanced colleagues correct me).

So, it is the values ​​of the imaginary part that determine the positions of prime numbers, and the famous ‘critical strip‘ in general, nothing.

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