[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.0021169954+pi+i%29%29+%2F+%28Zeta%281-x%2B2.0021169954+pi+i%29%29%29%29+%2B%281%2F236.30994+%28x-0.5%29%5E3%29+-1%2F30391.7%28x%29%28x-0.5%29%5E3%28x-1%29+%2B1%2F2150000%28x%29%28x-0.2%29%28x-0.5%29%28x-0.8%29%28x-1%29from+-0.01+to+1.01

This leaves residue of seventh order (septic equation).
Unfortunately the equation becomes too long for WolframAlpha to process. But this is how the septic equation looks like.

https://www.wolframalpha.com/input?i=plot+-1%2F2600000%28x%29%28x-0.07%29%28x-0.2%29%28x-0.5%29%28x-0.8%29%28x-0.93%29%28x-1%29from+-0.01+to+1.01

We seem to get a pattern here.
S(x) ≈ -ax^3 +bx^5 -cx^7 + ...

[hamdani yusuf (OP)]

We seem to get a pattern here.
S(x) ≈ -ax^3 +bx^5 -cx^7 + ...

This polynomial formula with alternating sign shouldn't be surprising. Riemann's Zeta function in critical strip is an analytic continuation from the original Zeta function which isn't convergent in that strip. This analytic continuation involves alternating sum.

The Riemann zeta function or Euler?Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as

for Re(𝑠)>1, and its analytic continuation elsewhere.
https://en.wikipedia.org/wiki/Riemann_zeta_function

The functional equation was established by Riemann in his 1859 paper "On the Number of Primes Less Than a Given Magnitude" and used to construct the analytic continuation in the first place. An equivalent relationship had been conjectured by Euler over a hundred years earlier, in 1749, for the Dirichlet eta function (the alternating zeta function):

Incidentally, this relation gives an equation for calculating ζ(s) in the region 0 < Re(s) < 1, i.e.

In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0:

This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ζ(s) ? and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ζ*(s). The following relation holds:

Both the Dirichlet eta function and the Riemann zeta function are special cases of polylogarithms.
https://en.wikipedia.org/wiki/Dirichlet_eta_function

[hamdani yusuf (OP)]

From the equation above, we can construct S function by taking the log of this function below.

The S function is the log of the ratio above.
https://en.wikipedia.org/wiki/Natural_logarithm#Series

[hamdani yusuf (OP)]

We seem to get a pattern here.
S(x) ≈ -ax^3 +bx^5 -cx^7 + ...

The summation seems to be missing the x^1 term. It turns out that it's because the imaginary part of the input has been selected to make it 0. Here's what we get when it's not the case.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.002117+pi+i%29%29+%2F+%28Zeta%281-x%2B2.002117+pi+i%29%29%29%29+%2B%281%2F236.30994+%28x-0.5%29%5E3%29+-1%2F30391.7%28x%29%28x-0.5%29%5E3%28x-1%29+%2B1%2F2150000%28x%29%28x-0.2%29%28x-0.5%29%28x-0.8%29%28x-1%29from+-0.1+to+1.1

And the result after the first order correction is applied.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.002117+pi+i%29%29+%2F+%28Zeta%281-x%2B2.002117+pi+i%29%29%29%29+%2B%281%2Fe%5E19.9%28x-0.5%29%29%2B%281%2F236.30994+%28x-0.5%29%5E3%29+-1%2F30391.7%28x%29%28x-0.5%29%5E3%28x-1%29+%2B1%2F2150000%28x%29%28x-0.2%29%28x-0.5%29%28x-0.8%29%28x-1%29from+-0.1+to+1.1

The results are different below the inflection point.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.002116+pi+i%29%29+%2F+%28Zeta%281-x%2B2.002116+pi+i%29%29%29%29+-%281%2Fe%5E14.5143%28x-0.5%29%29%2B%281%2F236.30994+%28x-0.5%29%5E3%29+-1%2F30391.7%28x%29%28x-0.5%29%5E3%28x-1%29+%2B1%2F2150000%28x%29%28x-0.2%29%28x-0.5%29%28x-0.8%29%28x-1%29from+-0.1+to+1.1

[hamdani yusuf (OP)]

https://www.popularmechanics.com/science/math/a61491065/mathematicians-close-to-solving-notorious-math-problem-riemann-hypothesis/

In the new paper, Maynard and Guth focus on a new limitation of Dirichlet polynomials. These are special series of complex numbers that many believe are of the same type as the function involved in the Riemann hypothesis involves. In the paper, they claim they?ve proven that these polynomials have a certain number of large values, or solutions, within a tighter range than before.

In other words, if we knew there might be an estimated three Dirichlet values between 50 and 100 before, now we may know that range to be between 60 and 90 instead. The eye exam just switched a blurry plate for a slightly less blurry one, but we still haven?t found the perfect prescription. ?If one knows some more structure about the set of large values of a Dirichlet polynomial, then one can hope to have improved bound,? Maynard and Guth conclude.

No, this is not a final proof of the Riemann hypothesis. But no one is suggesting it is. In advanced math, narrowing things down is also vital. Indeed, even finding out that a promising idea turns out to be wrong can have a lot of value?as it has a number of times in the related Twin Primes Conjecture that still eludes mathematicians.

In a collaboration that has lasted 160 years and counting, mathematicians continue to take each step together and then, hopefully, compare notes.

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2+pi+i%29%29+%2F+%28Zeta%281-x%2B2+pi+i%29%29%29%29-%281%2Fe%5E19.0501%28x-0.5%29%29%2B%281%2Fe%5E5.46372%28x%29%28x-0.5%29%28x-1%29%29-%281%2Fe%5E10.335%28x%29%28x-0.2%29%28x-0.5%29%28x-0.8%29%28x-1%29%29+from+-0.2+TO+1.2
The coefficients are deliberately chosen to hit the x axis at 0, 0.5, and 1.

With lower imaginary part, we get different set of coefficients.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2Bpi+i%29%29+%2F+%28Zeta%281-x%2B+pi+i%29%29%29%29+-%281%2Fe%5E0.3663635%28x-0.5%29%29from+-0.1+to+1.1

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2Bpi+i%29%29+%2F+%28Zeta%281-x%2B+pi+i%29%29%29%29+-%281%2Fe%5E0.3663635%28x-0.5%29%29++%2B%281%2Fe%5E4.079%28x%29%28x-0.5%29%28x-1%29%29+++from+-0.5+to+1.5

[hamdani yusuf (OP)]

We seem to get a pattern here.
S(x) ≈ -ax^3 +bx^5 -cx^7 + ...

The summation seems to be missing the x^1 term. It turns out that it's because the imaginary part of the input has been selected to make it 0. Here's what we get when it's not the case.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.002117+pi+i%29%29+%2F+%28Zeta%281-x%2B2.002117+pi+i%29%29%29%29+%2B%281%2F236.30994+%28x-0.5%29%5E3%29+-1%2F30391.7%28x%29%28x-0.5%29%5E3%28x-1%29+%2B1%2F2150000%28x%29%28x-0.2%29%28x-0.5%29%28x-0.8%29%28x-1%29from+-0.1+to+1.1

And the result after the first order correction is applied.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.002117+pi+i%29%29+%2F+%28Zeta%281-x%2B2.002117+pi+i%29%29%29%29+%2B%281%2Fe%5E19.9%28x-0.5%29%29%2B%281%2F236.30994+%28x-0.5%29%5E3%29+-1%2F30391.7%28x%29%28x-0.5%29%5E3%28x-1%29+%2B1%2F2150000%28x%29%28x-0.2%29%28x-0.5%29%28x-0.8%29%28x-1%29from+-0.1+to+1.1

The results are different below the inflection point.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.002116+pi+i%29%29+%2F+%28Zeta%281-x%2B2.002116+pi+i%29%29%29%29+-%281%2Fe%5E14.5143%28x-0.5%29%29%2B%281%2F236.30994+%28x-0.5%29%5E3%29+-1%2F30391.7%28x%29%28x-0.5%29%5E3%28x-1%29+%2B1%2F2150000%28x%29%28x-0.2%29%28x-0.5%29%28x-0.8%29%28x-1%29from+-0.1+to+1.1

The equations were obtained through visually approximate the position of the roots, which were identified by where the plot crosses the horizontal axis.

https://www.wolframalpha.com/input?i=y%3D%281%2Fe%5E19.9%28x-0.5%29%29%2B%281%2F236.30994+%28x-0.5%29%5E3%29+-1%2F30391.7%28x%29%28x-0.5%29%5E3%28x-1%29+%2B1%2F2150000%28x%29%28x-0.2%29%28x-0.5%29%28x-0.8%29%28x-1%29
When expanded, this is what we get.
0.0000324386 x^5 - 0.0000810965 x^4 - 0.0041587 x^3 + 0.00631915 x^2 - 0.00316972 x + y + 0.000528967 = 0
The visual approach only gives odd powered variable, but the fully expanded polynomial equation shows alternating sign between odd and even powered variable, just like Dirichlet's η function.

[hamdani yusuf (OP)]

The plot below shows the value of S function along the imaginary axis, overlaid with an approximation by a logarithmic function.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x+i%29%29+%2F+%28Zeta%281%2Bx++i%29%29%29%29+++%2B+i%281%2F2+ln+%28%7Cx%7C%29%29+-i+0.9189385332+from+-10+TO+10

There is significant deviation near ix = 0.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x+i%29%29+%2F+%28Zeta%281%2Bx++i%29%29%29%29+++%2B+i%281%2F2+ln+%28%7Cx%7C%29%29+-i+0.9189385332+from+-1+TO+1

But the deviation diminishes at higher imaginary value.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x+i%29%29+%2F+%28Zeta%281%2Bx++i%29%29%29%29+++%2B+i%281%2F2+ln+%28%7Cx%7C%29%29+-i+0.9189385332+from+0+TO+1000000

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x+i%29%29+%2F+%28Zeta%281%2Bx++i%29%29%29%29+++%2B+i%281%2F2+ln+%28%7Cx%7C%29%29+-i+0.9189385332+from++1000000+to+1000000.1

[hamdani yusuf (OP)]

I think I am getting closer to a proof of Riemann Hypothesis. There might be more proof will be found in the future. But the first one is the most important.

At least initially, this research involved a lot of curve fittings with formulas. It's commonly found in physics, such as in the development of Keppler's laws, Balmer's series, and Planck's formula. But it's less common in mathematics. I found it useful for recognizing the pattern, so I can decide if a direction of exploration worth pursuing any further. The final form of the proof still better be analytical through algebraic equations.

[hamdani yusuf (OP)]

Zeros of Zeta function seems to coincide with zeros of Dirichlet's eta function.
https://www.wolframalpha.com/input?i=plot+abs%28dirichlet+eta%280.5%2Bsi%29%29+from+0+to+30

https://www.wolframalpha.com/input?i=plot+log%28dirichlet+eta%28s%29%29+from+-10+to+1

[hamdani yusuf (OP)]

Further study on Dirichlet's Eta function using spreadsheet has given me insight to its behavior.


Here's the result on the first non-trivial zero of Zeta function. The horizontal axis shows the value of n in the summation formula.


When the imaginary part is changed.


When the real part is changed instead

[hamdani yusuf (OP)]

S function is visually more convincing in demonstrating that at high imaginary part, Zeta function on the right side of critical line must be closer to zero compared to its left side counterpart (its reflection about the critical line). But log function in the complete formula of the S function is not necessary to prove Riemann's Hypothesis. It complicates the formula by making it a double summation. The inequality still applies even without the log function.
|ζ(s)|< |ζ(1-s)| for Re(s)>0.5

[hamdani yusuf (OP)]

The most beautiful equation in math, explained visually [Euler?s Formula]

Calculating Riemann Zeta function requires exponentiation by complex numbers.

[hamdani yusuf (OP)]

Further study on Dirichlet's Eta function using spreadsheet has given me insight to its behavior.


[/quote]
Here's the plot to show a few smallest non-trivial zeros of Zeta function.
https://www.wolframalpha.com/input?i=plot+zeta+%280.5+%2B+i+s%29+from+0+to+30

Compare with the plot to show a few smallest zeros of Eta function along the critical line.
https://www.wolframalpha.com/input?i=plot+dirichlet+eta+%280.5+%2B+i+s%29+from+0+to+30

Adding them both gives the same positions for zeros.
https://www.wolframalpha.com/input?i=plot+dirichlet+eta+%280.5+%2B+i+s%29%2Bzeta+%280.5+%2B+i+s%29+from+0+to+30


Screenshot 2025-04-24 152143.png (32.08 kB . 414x336 - viewed 436 times)

[hamdani yusuf (OP)]

When the Eta function is divided by the Zeta function, we get a periodic curve in complex plane.
https://www.wolframalpha.com/input?i=plot+dirichlet+eta+%280.5+%2B+i+s%29%2Fzeta+%280.5+%2B+i+s%29+from+0+to+30

This is the same as ratio between Eta and Zeta function.
https://www.wolframalpha.com/input?i=plot+%28%281-2%5E%280.5-i+s%29%29%29+from+0+to+30

When the ratio is inverted, we get another periodic curve in complex plane.
https://www.wolframalpha.com/input?i=plot+zeta+%280.5+%2B+i+s%29+%2Fdirichlet+eta+%280.5+%2B+i+s%29+from+0+to+30

https://www.wolframalpha.com/input?i=plot+%281%2F%281-2%5E%280.5-i+s%29%29%29+from+0+to+30

Note that the ratio never crosses zero, hence its inverse never crosses infinity, and vice versa.

https://www.wolframalpha.com/input?i=plot+abs%28dirichlet+eta+%280.5+%2B+i+s%29%2Fzeta+%280.5+%2B+i+s%29%29+from+0+to+30

https://www.wolframalpha.com/input?i=plot+abs%281%2F%28dirichlet+eta+%280.5+%2B+i+s%29%2Fzeta+%280.5+%2B+i+s%29%29%29+from+0+to+30

[hamdani yusuf (OP)]

Note that the ratio never crosses zero, hence its inverse never crosses infinity, and vice versa.

But we know that Riemann zeta function does produce zeros in critical line. It means that they get zeros from the same positions.

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+%28%2Bzeta%28s%2Bi%29-zeta%281-s-i%29+%29from+0+to+1
(+zeta(s+i) -zeta(1-s-i) ) from 0 to 1
The colored operators determine whether the resulting curves are odd or even function.

In the image above, the real part is an odd function, which has rotational symmetry about point 0.5, while the imaginary part is an even function, which has reflectional symmetry about the critical line.

The table below shows all the possible combination of the operators and the resulting curves.


The functional equation and the behavior of complex conjugates in Zeta function guarantee that any non-trivial zero of Zeta function yields zero for all 16 possible combination of operators as shown in the table.

[hamdani yusuf (OP)]

Here's what they look like for the first non-trivial zero
https://www.wolframalpha.com/input?i=plot+%28%2Bzeta%28s%2Bi+14.1347%29-zeta%281-s-i+14.1347%29+%29from+0+to+1

https://www.wolframalpha.com/input?i=plot+%28%2Bzeta%28s%2Bi+14.1347%29%2Bzeta%281-s-i+14.1347%29+%29from+0+to+1

https://www.wolframalpha.com/input?i=plot+%28%2Bzeta%28s%2Bi+14.1347%29%2Bzeta%281-s%2Bi+14.1347%29+%29from+0+to+1

[hamdani yusuf (OP)]

For any non-trivial zeros of zeta function all of the even curves must intersect at the same locations.
https://www.wolframalpha.com/input?i=plot+%28%2Bzeta%28s%2B14i%29%2Bzeta%281-s%2B14i%29+%29from+0+to+1
Meanwhile, the odd curves always intersect at 0.5.
https://www.wolframalpha.com/input?i=plot+%28%2Bzeta%28s%2B14i%29-zeta%281-s%2B14i%29+%29from+0+to+1

[hamdani yusuf (OP)]

If there any violation to Riemann hypothesis, the odd curves must intersect at zero more than once.
https://www.wolframalpha.com/input?i2d=true&i=plot+%5C%2840%29%5C%2840%29zeta%5C%2840%29x%2B16.17842i%5C%2841%29%5C%2841%29-%5C%2840%29Zeta%5C%2840%291-x%2B16.17842i%5C%2841%29%5C%2841%29%5C%2841%29from+-0.01to+1.01

https://www.wolframalpha.com/input?i2d=true&i=plot+re+%5C%2840%29%5C%2840%29zeta%5C%2840%29x%2B16.17842i%5C%2841%29%5C%2841%29-%5C%2840%29Zeta%5C%2840%291-x%2B16.17842i%5C%2841%29%5C%2841%29%5C%2841%29from+-0.01to+1.01

https://www.wolframalpha.com/input?i2d=true&i=plot+re+%5C%2840%29%5C%2840%29zeta%5C%2840%29x%2B16.16842i%5C%2841%29%5C%2841%29-%5C%2840%29Zeta%5C%2840%291-x%2B16.16842i%5C%2841%29%5C%2841%29%5C%2841%29from+0.3+to+0.7

https://www.wolframalpha.com/input?i2d=true&i=plot++%5C%2840%29%5C%2840%29zeta%5C%2840%29x%2B16.16842i%5C%2841%29%5C%2841%29-%5C%2840%29Zeta%5C%2840%291-x%2B16.16842i%5C%2841%29%5C%2841%29%5C%2841%29from+0.3+to+0.7

Which in the examples above is not the case.
At the same time, the even functions must also intersect at the same positions.

https://www.wolframalpha.com/input?i2d=true&i=plot++%5C%2840%29%5C%2840%29zeta%5C%2840%29x%2B16.16842i%5C%2841%29%5C%2841%29%2B%5C%2840%29Zeta%5C%2840%291-x%2B16.16842i%5C%2841%29%5C%2841%29%5C%2841%29from+0.3+to+0.7

https://www.wolframalpha.com/input?i2d=true&i=plot++%5C%2840%29%5C%2840%29zeta%5C%2840%29x-16.16842i%5C%2841%29%5C%2841%29%2B%5C%2840%29Zeta%5C%2840%291-x-16.16842i%5C%2841%29%5C%2841%29%5C%2841%29from+0.3+to+0.7