Applied Science and Innovative Research ISSN 2474-4972 (Print) ISSN 2474-4980 (Online) Vol. 6, No. 1, 2022 www.scholink.org/ojs/index.php/asir 14 Original Paper Riemann’s Hypothesis New Proof Fausto Galetto1 1 Independent researcher, past professor of Quality Management at Politecnico of Turin, Turin, Italy Received: December 28, 2021 Accepted: January 12, 2022 Online Published: January 19, 2022 doi:10.22158/asir.v6n1p14 URL: http://doi.org/10.22158/asir.v6n1p14 Abstract After some papers proving the RC (short for “Riemann’s Conjecture”, also known as the “Riemann’s Hypothesis”, RH), now the author provides a new proof, using the “Spira Criterion” that states “The RH is equivalent to the statement that if >0.5 and t> 6.5 then |(1-s)|> |(s)|”. We use the concept of “transfer function” for control systems. This new proof is so simple that the author wonders why a great mathematician like Riemann did not see it; therefore F. Galetto thinks that somewhere in the purported proof there should be an error. Keywords Zeta Function, Functional Equation, Nontrivial Zeros, Critical Line, Spira Criterion, Generalised Power Transfer Function. 1. Introduction It is well known that for over a century mathematicians have been trying to prove the so-called Riemann Hypothesis, RH for short, a conjecture claimed by Riemann [who was professor at University of Gottingen in Germany], near 1859 in a 8-page paper “On the number of primes less than a given magnitude” shown at Berlin Academy, and dated/published in 1859; it is well known, as well, that RH is related to set of all the Prime Numbers (Figure 1); prime numbers are fundamental for encryption of documents and data (e.g. payments with credit cards). He invented the Riemann zeta function: (z), where z is a complex number z=x+iy and i is the “positive” imaginary unit such that i2=-1. In the following part of the paper we will use the symbols s=+it for the complex variable, as in Titchmarsh (1986). www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 6, No. 1, 2022 15 Published by SCHOLINK INC. …………… Figure 1. From the Original B. Riemann Manuscript The description the zeta function (𝑠) = (𝜎 + 𝑖𝑡) will not be introduced here, interested readers can refer to any text book, e.g., Titchmarsh (1986), Bombieri (2000), Broughan (2017). (𝑠) is a meromorphic function with only a simple pole at s=1 [with residue 1]; moreover (𝑠)0 for all z  C with Re(s)=>1; it has zeros at the negative even integers -2, -4, -6, ….., named trivial zeros. The other zeros are named nontrivial zeros: all the “known” zeros, computed up to now [up to 2004, 1012 zeros have been computed, all on the Critical Line], are the complex numbers s=1/2 + it, with suitable values of t. It satisfies the functional equation π− 𝑠 2 Γ ( 𝑠 2 ) (s) = π−(1−𝑠)/2 Γ[(1 − 𝑠)/2])(1 − s) (1) The properties of (s) are as follows: a) (s) has no zero for Re(s)>1; b) the only pole (s) is at s=1: it is simple and has residue 1; c) (s) has trivial zeros at the negative even integers z = -2, -4, -6, …. d) all the nontrivial zeros lie inside the region, named Critical Strip, 0  Re(s)  1 and are symmetric about both the vertical line, named Critical Line, Re(z)==1/2 and the real axis Im(z)=0: (𝑠) = (𝑠)̅̅ ̅̅ ̅̅ = (�̅�) = (1 − 𝑠) = (1 − �̅�) Riemann conjectured the so-called Riemann Hypothesis (RH), or Conjecture (RC): RC states All nontrivial zeros of (s) have real part  equal to 1/2 . Many great mathematicians tackled this problem; we do not mention them, because they can be found in many books and papers. If RC would be related to Physics, it would be considered a “universal law”: up to 2004, 1012 zeros have been computed, all on the Critical Line. www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 6, No. 1, 2022 16 Published by SCHOLINK INC. If RC would be related to Statistics, <>, would be confirmed with a Confidence Level (CL) > 0.9999999999: the evidence of 1012 zeros computed, all on the Critical Line (as to 2004) supports H0 with that “high” CL. But Mathematics asks much more than Statistics and Physics… Also a theorem of G. Hardy [Hardy’s Theorem, 1914] that proved that “There are infinitely many zeros of (s) on the Critical Line” is not enough. ALL the nontrivial zeros must be on the Critical Line, if one wants to prove RC. The author, Fausto Galetto, is aware that (in these weeks) he has been affording a very important problem that great mathematicians have failed to prove. 2. The “Transfer Function” and the Spira Criterion From (1) we derive (1 − s) = π0.5−𝑠 Γ(𝑠/2) Γ[(1−𝑠)/2]) (s) (2a) which can be written, in the form used in electronics, communication theory and control systems, (1 − s) = H(s)(s) (2b) where H(s) is the “Transfer Function” that links the Input (s) to the Output (1 − s). The “Transfer Function” is the ratio Output/Input. When =0 the “Transfer Function” depends only on the “frequency” ; from now on we use the symbol s=+j, as it is customary in electronics, communication theory and control systems: (1 − σ − jω) = H(σ + jω)(σ + jω) (2c) When =0 we have (1 − jω) = H(jω)(jω) (2d) Squaring both side and taking the absolute value, we get |(1 − jω)|2 = |H(jω)|2|(jω)|2 (3) This formula is the same as the following S𝑂𝑢𝑡𝑝𝑢𝑡(jω) = |H(jω)|2S𝐼𝑛𝑝𝑢𝑡(jω) (3b) used in electronics, communication theory and control systems, where the function S(j) is the “Power Spectral Density”. We name here |H(jω)|2 the “Power Transfer Function”. In analogy to (3b) we define |(1 − s)|2 = |H(s)|2|(s)|2 (3c) and name here |H(s)|2 the “Generalised Power Transfer Function”. The Spira Criterion (Broughan, 2017) for RC is The Riemann Hypothesis is equivalent to the statement that |(1 − 𝑠)| > |(𝑠)| for any 0.5<<1 and any 𝜔 ≥ 2𝜋 + 0.1. Notice that |H(s)| > 0 for any s and |H(0.5 + jω)| = 1 because |H(1 − s)H(s)| = 1 www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 6, No. 1, 2022 17 Published by SCHOLINK INC. Since |(1 − 𝑠)/(𝑠)| = |H(s)|, the Spira Criterion (Broughan, 2017) for RC states that |H(σ + jω)| > 1 for any 0.5<<1 and any 𝜔 ≥ 2𝜋 + 0.1. We can use the infinite product expansion for the Gamma Function and we get |H(s)| = |𝜋0.5−𝑠| [ |Γ( 𝑠 2 )| |Γ( 1−𝑠 2 )| ] = |𝜋0.5−𝑠||𝑒𝛾(0.5−𝑠)| | 1−𝑠 𝑠 |∏ { |(1+ 1−𝑠 2𝑛 )| |(1+ 𝑠 2𝑛 )| |𝑒 𝑠−0.5 𝑛 |}∞ 1 = |𝜋0.5−𝑠||𝑒𝛾(0.5−𝑠)| | 1−𝑠 𝑠 |∏ { |(2𝑛+1−𝑠)| |(2𝑛+𝑠)| |𝑒 𝑠−0.5 𝑛 |}∞ 1 (4) Figure 2 depicts one of the “products” within the braces Figure 2. Picture of One of the “Products” within the Braces We can also find the “Generalised Power Transfer Function” |H(s)|2 = |𝜋0.5−𝑠|2|[𝑒−𝛾(𝑠−0.5)]| 2 | 1−𝑠 𝑠 | 2 ∏ { |(2𝑛+1−𝑠)|2 |(2𝑛+𝑠)|2 |𝑒 𝑠−0.5 𝑛 | 2 }∞ 1 (5) This can be written as a real function A(,) A(σ,ω) = 𝜋1−2𝜎𝑒−𝛾(2𝜎−1) (1−𝜎)2+𝜔2 𝜎2+𝜔2 ∏ { (2𝑛+1−𝜎)2+𝜔2 (2𝑛+𝜎)2+𝜔2 𝑒 (2𝜎−1) 𝑛 }∞ 1 (5b) 3. The Proof of RC To prove RC, we use the Spira Criterion (Broughan, 2017) for RC: we show that |H(s)| = |H(σ + jω)| > 1 for any 0.5<<1 and any ω ≥ 2π + 0.1. For any chosen 𝜔 ≥ 2𝜋 + 0.1, we can find a value n0 such that, for any n>n0, the logarithm is expended as ln [1 + 𝑧/(2𝑛)] ≈ {𝑧/(2𝑛) − 𝑧2/[2(2𝑛)2]}, with 𝑧 = σ + jω; then 𝑙𝑛 (1 + 1 − 𝑠 2𝑛 ) − 𝑙𝑛 (1 + 𝑠 2𝑛 ) ≈ (1 − 2𝑠) 2𝑛 [1 − 1 2(2𝑛) ] Therefore, for any n>n0, each factor in the “infinite product” is 𝑎𝑛 = { |1 + (1 − 𝑠)/(2𝑛)| |1 + 𝑠/(2𝑛)| |𝑒 𝑠−0.5 𝑛 |} ≈ |𝑒 (𝑠−0.5)[ 1 (2𝑛)2 ] | = 𝑒 𝑅𝑒{(𝑠−0.5)[ 1 (2𝑛)2 ]} = 𝑒 (𝜎−0.5)[ 1 (2𝑛)2 ] > 1 Points of the ZETA function 0 0.5 1 Re(s) Im(s) 1-s -2n 2n+s 2n+1-s s www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 6, No. 1, 2022 18 Published by SCHOLINK INC. See Figure 3 Figure 3. Picture of the First Factor within the Braces Since 𝑎𝑛 > 𝑎𝑛+1, we get ∏ 𝑎𝑛 ∞ 1 > 1 and hence |H(s)| > 1. Therefore RC is proved by the Spira Criterion (Broughan, 2017). Figure 4. Behaviour of |𝐇(𝛔 + 𝐣𝛚)|, Abscissa is 𝛔, Curves are Indexed by 𝛚 0 0.5 1 Re(s) Im(s) 1-s s 1+s/2s/2 (1-s)/2 1.5 s-0.5 + − + − . For n=1 www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 6, No. 1, 2022 19 Published by SCHOLINK INC. Figure 4 show the behaviour of |H(σ + jω)| for various values of (abscissa) σ and various ω (curves); all curves intersects at σ=0.5 and have ordinate |H(σ + jω)| = 1 We can confirm the previous result by writing it in a different way (Abramowitz et al., 1965) the gamma function; so the absolute “transfer function” becomes |H(s)| = |𝜋0.5−𝑠| |Γ( 𝜎 2 )| |Γ( 1−𝜎 2 )| √∏ [1 + ( 𝜔/2 𝑛+(1−𝜎)/2 ) 2 ] / [1 + ( 𝜔/2 𝑛+𝜎/2 ) 2 ]∞ 0 (4b) Let’s now consider the “Generalised Power Transfer Function” |H(s)|2; from above, we have the power difference |(1 − s)|2 − |(s)|2 = |(s)|2{|H(s)|2 − 1} > 0. Let’s consider the equation |(s)|2{|H(s)|2 − 1} = 0 (6) Form (6) we find two solutions |(s)|2 = 0 (7) {|H(s)|2 − 1} = 0 (8) (7) entail that s = σ + jω is a zero of the Zeta Function (σ + jω), while (8) entails that the real function A(,) equals 1 for any  𝜋1−2𝜎𝑒−𝛾(2𝜎−1) (1−𝜎)2+𝜔2 𝜎2+𝜔2 ∏ { (2𝑛+1−𝜎)2+𝜔2 (2𝑛+𝜎)2+𝜔2 𝑒 (2𝜎−1) 𝑛 }∞ 1 = 1 (9) It is easily seen that (9) has a unique solution 𝜎 = 0.5; therefore the zeros of the Zeta Function (σ + jω) are all on the Critical Line 𝑠 = 0.5 + 𝑗𝜔; this is the same result found before by the Spira Criterion. We can confirm the previous result by writing in a different way (Abramowitz et al., 1965) the gamma function; so the “Generalised Power Transfer Function” |H(s)|2 becomes |H(s)|2 = 𝜋1−2𝜎 [ |Γ( 𝜎 2 )| |Γ( 1−𝜎 2 )| ] 2 ∏ [1 + ( 𝜔/2 𝑛+(1−𝜎)/2 ) 2 ] / [1 + ( 𝜔/2 𝑛+𝜎/2 ) 2 ]∞ 0 = 1 (9b) It is easily seen that (9b) has a unique solution 𝜎 = 0.5; therefore …. (as before). 4. Conclusion Titchmarsh proved that there are infinite zeros of the Riemann zeta function (s) in the Critical Strip, that is there are infinite values sk=k+ik such that (sk)=0, [0 < k < 1]. G. Hardy [Hardy’s Theorem, 1914] proved that “There are infinitely many zeros of (s) on the Critical Line”: that fact was not conclusive, because ALL the zeros must be on the Critical Line. In this paper, to prove RC we used first the “Spira Criterion” [with the Transfer Function H(s)] and second the “Generalised Power Transfer Function” |H(s)|2. The result is that RH (RC) is true. (as done in previous papers, Fausto Galetto, 2014, 2018, 2019). www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 6, No. 1, 2022 20 Published by SCHOLINK INC. References Abramowitz, M., & Stegun, I. A. (1965). Handbook of Mathematical Functions. National Bureau of Standards, Applied Math. Series #55, Dover Publications, sec. 6.5. Bombieri, E. (2000). 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