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2.5: The Riemann Hypothesis

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• J. J. P. Veerman
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Definition 2.19

The Riemann zeta function \(\zeta(z)\) is a complex function defined as follows on \(\{z \in \mathbb{C} | \mbox{Re}z > 1\}\)

\[\zeta (z) = \sum_{n=1}^{\infty} n^{-z} \nonumber\]

On other values of \(z \in \mathbb{C}\) it is defined by the analytic continuation of this function (except at \(z = 1\) where it has a simple pole).

Analytic continuation is akin to replacing \(e^x\) where \(x\) is real by \(e^z\) where \(z\) is complex. Another example is the series \(\sum_{j=0}^{\infty} z^j\). This series diverges for \(|z| > 1\). But as an analytic function, it can be replaced by \((1-z)^{-1}\) on all of \(\mathbb{C}\) except at the pole \(z = 1\) where it diverges.

Recall that an analytic function is a function that is differentiable. Equivalently, it is a function that is locally given by a convergent power series. If \(f\) and \(g\) are two analytic continuations to a region \(U\) of a function \(h\) given on a region \(V \subset U\), then the difference \(f-g\) is zero on some \(U\) and therefore all its power expansions are zero and so it must be zero on the the entire region. Hence, analytic conjugations are unique. That is the reason they are meaningful. For more details, see for example [ 4 , 14 ].

It is customary to denote the argument of the zeta function by \(s\). We will do so from here on out. Note that \(|n-s| = n-\mbox{Re} s\), and so for \(\mbox{Re} s > 1\) the series is absolutely convergent. At this point, the student should remember – or look up in [ 23 ] – the fact that absolutely convergent series can be rearranged arbitrarily without changing the sum. This leads to the following proposition.

Proposition 2.20

For \(\mbox{Re} s > 1\) we have

\[\sum_{n=1}^{\infty} n-s = \prod_{p prime} (1-p^{-s})^{-1} \nonumber\]

There are two common proofs of this formula. It is worth presenting both.

The first proof uses the Fundamental Theorem of Arithmetic. First, we recall that use geometric series

\[(1-p^{-s})^{-1} = \sum_{k=0}^{\infty} p^{-ks} \nonumber\]

to rewrite the right hand of the Euler product. This gives

\[\prod_{p prime} (1-p^{-s})^{-1} = (\sum_{k_1 = 0}^{\infty} p_{1}^{-k_{1}s}) (\sum_{k_2 = 0}^{\infty} p_{2}^{-k_{2}s}) (\sum_{k_3 = 0}^{\infty} p_{3}^{-k_{3}s}) \nonumber\]

Re-arranging terms yields

\[\dots = (p_{1}^{k_{1}}p_{2}^{k_{2}}p_{3}^{k_{3}} \dots)^{-s} \nonumber\]

By the Fundamental Theorem of Arithmetic, the expression \((p_{1}^{k_{1}}p_{2}^{k_{2}}p_{3}^{k_{3}} \dots)\) runs through all positive integers exactly once. Thus upon re-arranging again we obtain the left hand of Euler’s formula.

The second proof, the one that Euler used, employs a sieve method. This time, we start with the left hand of the Euler product. If we multiply \(\zeta\) by \(2^{-s}\), we get back precisely the terms with \(n\) even. So

\[(1-2^{-s}) \zeta(s) = 1+3^{-s}+5^{-s}+\cdots = \sum_{2 \nmid n} n^{-s} \nonumber\]

Subsequently we multiply this expression by \((1-3^{-s})\). This has the effect of removing the terms that remain where \(n\) is a multiple of \(3\). It follows that eventually

\[(1-p_{l}^{-s}) \dots (1-p_{1}^{-s}) \zeta (s) = \sum_{p_{1} \nmid n, \cdots p_{l} \nmid n} n^{-s} \nonumber\]

The argument used in Eratosthenes sieve (Section 1.1) now serves to show that in the right hand side of the last equation all terms other than \(1\) disappear as l tends to infinity. Therefore, the left hand tends to 1, which implies the proposition.

The most important theorem concerning primes is probably the following (without proof).

Figure 3. On the left, the function \(\int_{2}^{x} \mbox{ln} t dt\) in blue, \(\pi (x)\) in red, and \(x/ \mbox{ln}x\) in green. On the right, we have \(\int_{2}^{x} \mbox{ln} t dt - x/\mbox{ln} x\) in blue, \(\pi (x)- x/\mbox{ln} x\) in red.

Theorem 2.21 (Prime Number Theorem)

Let \(\pi (x)\) denote the prime counting function, that is: the number of primes less than or equal to \(x > 2\).

• \(\lim_{x \rightarrow \infty} \frac{\pi (x)}{(x/\mbox{ln} x)} = 1\) and
• \(\lim_{x \rightarrow \infty} \frac{\pi (x)}{\int_{2}^{x} \mbox{ln} t dt} = 1\)

where \(\mbox{ln}\) is the natural logarithm.

The first estimate is the easier one to prove, the second is the more accurate one. In Figure 3 on the left, we plotted, for \(x \in [2,1000]\), from top to bottom the functions \(\int_{2}^{x} \mbox{ln} t dt\) in blue, \(\pi (x)\) in red, and \(x/\mbox{ln} x\). In the right hand figure, we augment the domain to \(x \in [2, 100000]\). and plot the difference of these functions with \(x/\mbox{ln} x\). It now becomes clear that \(\int_{2}^{x} \mbox{ln} t dt\) is indeed a much better approximation of \(\pi (x)\). From this figure one may be tempted to conclude that \(\int_{2}^{x} \mbox{ln} t dt - \pi (x)\) is always greater than or equal to zero. This, however, is false. It is known that there are infinitely many n for which \(\int_{2}^{x} \mbox{ln} t dt - \pi (x) < 0\). The first such \(n\) is called the Skewes number. Not much is known about this number, except that it is less than 10317.

Perhaps the most important open problem in all of mathematics is the following. It concerns the analytic continuation of \(\zeta (s)\) given above.

Conjecture 2.22 (Riemann Hypothesis)

All non-real zeros of \(\zeta (s)\) lie on the line \(\mbox{Re} s = \frac{1}{2}\)

In his only paper on number theory [ 20 ], Riemann realized that the hypothesis enabled him to describe detailed properties of the distribution of primes in terms of of the location of the non-real zero of \(\zeta ( s )\). This completely unexpected connection between so disparate fields – analytic functions and primes in \(\mathbb{N}-\)spoke to the imagination and led to an enormous interest in the subject. In further research, it has been shown that the hypothesis is also related to other areas of mathematics, such as, for example, the spacings between eigenvalues of random Hermitian matrices [ 2 ], and even physics [ 5 , 6 ].

Riemann Hypothesis

A more general statement known as the generalized Riemann hypothesis conjectures that neither the Riemann zeta function nor any Dirichlet L-series has a zero with real part larger than 1/2.

Legend holds that the copy of Riemann's collected works found in Hurwitz's library after his death would automatically fall open to the page on which the Riemann hypothesis was stated (Edwards 2001, p. ix).

Proof of the Riemann hypothesis is number 8 of Hilbert's problems and number 1 of Smale's problems .

In 2000, the Clay Mathematics Institute ( http://www.claymath.org/ ) offered a $1 million prize ( http://www.claymath.org/millennium/Rules_etc/ ) for proof of the Riemann hypothesis. Interestingly, disproof of the Riemann hypothesis (e.g., by using a computer to actually find a zero off the critical line ), does not earn the $1 million award.

The Riemann hypothesis is equivalent to the statement that all the zeros of the Dirichlet eta function (a.k.a. the alternating zeta function)

By modifying a criterion of Robin (1984), Lagarias (2000) showed that the Riemann hypothesis is equivalent to the statement that

There is also a finite analog of the Riemann hypothesis concerning the location of zeros for function fields defined by equations such as

According to Fields medalist Enrico Bombieri, "The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers" (Havil 2003, p. 205).

In Ron Howard's 2001 film A Beautiful Mind , John Nash (played by Russell Crowe) is hindered in his attempts to solve the Riemann hypothesis by the medication he is taking to treat his schizophrenia.

In the Season 1 episode " Prime Suspect " (2005) of the television crime drama NUMB3RS , math genius Charlie Eppes realizes that character Ethan's daughter has been kidnapped because he is close to solving the Riemann hypothesis, which allegedly would allow the perpetrators to break essentially all internet security.

In the novel Life After Genius (Jacoby 2008), the main character Theodore "Mead" Fegley (who is only 18 and a college senior) tries to prove the Riemann Hypothesis for his senior year research project. He also uses a Cray Supercomputer to calculate several billion zeroes of the Riemann zeta function. In several dream sequences within the book, Mead has conversations with Bernhard Riemann about the problem and mathematics in general.

Portions of this entry contributed by Len Goodman

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Home — Millennium Problems — Riemann Hypothesis

Riemann Hypothesis

The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann’s 1859 paper, it asserts that all the ‘non-obvious’ zeros of the zeta function are complex numbers with real part 1/2.

Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called  prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern.  However, the German mathematician G.F.B. Riemann (1826 – 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function  ζ(s) = 1 + 1/2 s  + 1/3 s  + 1/4 s + …  called the  Riemann Zeta function . The Riemann hypothesis asserts that all  interesting solutions of the equation ζ(s) = 0 lie on a certain vertical straight line.

This has been checked for the first 10,000,000,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.

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Sketches of Physics pp 191–263 Cite as

A Primer on the Riemann Hypothesis

• Michael E. N. Tschaffon 21 ,
• Iva Tkáčová 22 ,
• Helmut Maier 23 &
• Wolfgang P. Schleich 24 , 25  
• First Online: 15 September 2023

385 Accesses

Part of the book series: Lecture Notes in Physics ((LNP,volume 1000))

We provide an introduction for physicists into the Riemann Hypothesis. For this purpose, we first introduce, and then compare and contrast the Riemann function and the Dirichlet L-functions, with the Titchmarsh counterexample. Whereas the first two classes of functions are expected to satisfy the Riemann Hypothesis, the Titchmarsh counterexample is known to violate it. Throughout our article we employ elementary mathematical techniques known to every physicist. Needless to say, we do not verify the Riemann Hypothesis but suggest heuristic arguments in favor of it. We also build a bridge to quantum mechanics by interpreting the Dirichlet series central to this field as a superposition of probability amplitudes leading us to an unusual potential with a logarithmic energy spectrum opening the possibility of factoring numbers.

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Acknowledgements

We thank P. C. Abbott, M. B. Kim, H. L. Montgomery, J. W. Neuberger, M. Zimmermann for many fruitful discussions, E. P. Glasbrenner for technical assistance and J. Pohl for her help with the scans of Riemann’s article. Moreover, we are most grateful to the Niedersächsische Staats- und Universitätsbibliothek and, in particular, R. B. Röper for allowing us to present excerpts of Riemann’s original manuscript. W. P. S. is also grateful to Texas A & M University for a Faculty Fellowship at the Hagler Institute for Advanced Study at Texas A& M University and to Texas A& M AgriLife for the support of this work. The research of the IQST is financially supported by the Ministry of Science, Research and Arts Baden-Württemberg. I. B. is grateful for the financial support from the Technical University of Ostrava, Grant No. SP2018/44.

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Institut für Quantenphysik and Center for Integrated Quantum Science and Technology (IQST), Universität Ulm, Ulm, Germany

Michael E. N. Tschaffon

Department of Physics, Faculty of Electrical Engineering and Computer Science, VSB-Technical University of Ostrava, Ostrava, Poruba, Czech Republic

Iva Tkáčová

Institut für Reine Mathematik, Universität Ulm, Ulm, Germany

Helmut Maier

Wolfgang P. Schleich

Hagler Institute for Advanced Study, Institute for Quantum Science and Engineering (IQSE), and Texas A&M AgriLife Research, Texas A&M University, College Station, TX, USA

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Maciej Lewenstein

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Angel Rubio

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Appendix 1: Functional Equation of Jacobi Theta Function

In this Appendix we briefly rederive the functional equation, Eq. ( 26 ), of the Jacobi theta function

defined by Eq. ( 21 ), in order to bring out most clearly the origin of the difference to the corresponding relation, Eq. ( 155 ), for the generalized Jacobi theta function given by Eq. ( 154 ).

Since the summation index n appears in \(\omega \) as a square, we can immediately include negative values of n together with a factor of \(1/2\) . However, the term corresponding to \(n=0\) is unity. Thus, we have to subtract this contribution, and arrive at the representation

of \(\omega \) .

Next we employ the Poisson summation formula

where \(b=b(\nu )\) is a continuous extension of the discrete coefficients \(b_n\) . We emphasize that the specific form of the extension is of no consequence as long as \(b(n)\equiv b_n\) .

Indeed, this fact is a result of the identity

which is at the very heart of the Poisson summation formula.

For the choice

and with the help of the integral relation

we arrive at the identity

which leads us with the representation, Eq. ( 195 ), of \(\omega \) to

that is, the familiar functional equation

We emphasize that the second term in Eq. ( 202 ), which is independent of \(\omega \) , is a consequence of the absence of the term \(n=0\) in the definition, Eq. ( 194 ), of the Jacobi theta function \(\omega \) leading to the subtraction of unity in Eq. ( 195 ).

Appendix 2: Equivalent Condition for Non-trivial Zeros

In Sect. 2.4 we have derived the condition, Eq. ( 39 ) , for a zero on the critical line given by the identity of an integral

containing an oscillatory integrand with a Lorentzian. We devote this appendix to the derivation of an alternative but equivalent formulation by casting the integral into a different form.

In particular, we analyze the asymptotic behavior of J . For this purpose, we first derive an exact alternative expression for J , and then consider the limit \(\tau \rightarrow \infty \) .

We start by introducing in J the new integration variable \(y\equiv (1/2)\ln x\) , or \(x=\exp (2y)\) , which with the identity \(\mathrm {d}y=\mathrm {d}x/(2x)\) leads us to the form

Next, we obtain by integration by parts the equivalent representation

where we have used the fact that the boundary terms vanish, and find performing the differentiation

Finally, we introduce the new integration variable \(\theta \equiv \tau y\) , and arrive at the expression

When we recall from Eq. ( 21 ) the definition

of \(\omega \) , the differentiation yields the explicit formula

which decays for increasing \(\tau \) as \(1/ \tau ^2\) which is identical to the decrease of the Lorentzian on the right-hand side of Eq. ( 39 ).

When we now compare this representation of J to the initial one, Eq. ( 203 ), we find, apart from the decay with \(1/\tau ^2\) , three distinct features: (i) The oscillatory function in the integral is independent of \(\tau \) . (ii) We have a double-exponential decay, but (iii) this rapid decay only sets in for integration variables \(\theta > \tau /2\) .

Hence, the expression, Eq. ( 209 ), for J shows that the main contribution to this integral results from the decays characterized by the summation index n that are fast compared to the oscillation period, in complete agreement with the discussion of Sect. 2.4 .

Appendix 3: Exponential Product

In this appendix we cast the product

where q is an integer and \(\kappa =0,1\) , into an exponential form, which allows us to derive in a straight-forward way an asymptotic limit of \(\lambda \) . This term is not only a factor in the definition, Eq. ( 151 ), of the Dirichlet L-function \(\Lambda \) but also appears in the Riemann function \(\xi \) , defined by Eq. ( 40 ) for the special case \(q=1\) and \(\kappa =0\) .

We first derive a general expression for \(\lambda \) and then perform the asymptotic limit. Finally, we address the special case of the Riemann function.

1.1 General Expression

We start from the definition, Eq. ( 210 ), of \(\lambda \) in the exponential form

and recall the representation [ 17 ]

of the logarithm of the Gamma function with the remainder

which leads us for the argument \((s+\kappa )/2\) to the expression

Here we have introduced the abbreviation

and \(\bar {R}\) is defined by Eq. ( 213 ).

1.2 Asymptotic Limit

Next we consider the limit \(s \rightarrow \infty \) . In this case we find with the help of the expansion

the relation

Moreover, the generating function [ 1 ]

of the Bernoulli numbers \(B_n\) with

allows us to expand the remainder \(\bar {R}\) given by Eq. ( 213 ) into inverse powers of s with the leading contribution

Hence, we arrive in the lowest order of \(1/s\) at the asymptotic expression

for \(\mathcal {R}\) , which apart from the constant term decays as \(1/s\) .

1.3 Special Case: Riemann Function

Finally, we discuss the example \(q=1\) and \(\kappa =0\) corresponding to the Riemann function \(\xi \) . In this case, Eq. ( 214 ) reduces to

where \(\mathcal {R}\) given by Eq. ( 215 ) simplifies to

With the help of the asymptotic limit Eq. ( 221 ), we find the expression

for the remainder, which together with Eq. ( 224 ) is consistent with Eq. ( 222 ) for \(\kappa =0\) .

Appendix 4: Lines of Constant Height and Constant Phase

In Sect. 3.6.3 we have derived an asymptotic expression for \(\xi \) which arises solely from the product \((\wp /2)\pi ^{-s/2}\Gamma \left (s/2\right )\) since we have made the approximation \(\zeta =1\) . This formula is in terms of the complex variable s .

In this appendix, we first cast this expression in terms of amplitude and phase, and then consider two special cases. We conclude by verifying these approximations using the Cauchy-Riemann differential equations in amplitude and phase discussed in the Chapter “Insights into Complex Functions” of this volume.

1.1 Decomposition in Amplitude and Phase

We start by decomposing the approximation

of \(\alpha _{\xi }\) into its real and imaginary parts \(\Sigma \) and \(\Phi \) , that is,

determining the absolute value

and the phase \(\Phi \) of \(\xi \) . Here, we first derive general expressions for \(\Sigma \) and \(\Phi \) , and then consider the two extreme limits \(1<\sigma \ll \tau \) and \(0\leq \tau \ll \sigma \) .

With the identity

leading us to

we find the explicit expressions

We emphasize that the decomposition, Eq. ( 227 ), of the approximation, Eq. ( 226 ), of \(\alpha _{\xi }\) into \(\Sigma \) and \(\Phi \) given by Eqs. ( 231 ) and ( 232 ) is exact.

1.2 Special Limits

In order to gain insight into the dependence of \(\Sigma \) and \(\Phi \) on \(\sigma \) and \(\tau \) , we now consider two extreme limits, and derive approximate but analytic expressions for \(\Sigma \) and \(\Phi \) .

1.2.1 At the Center of the Complex Plane

We start with the case of \(1<\sigma \ll \tau \) , and arrive with the help of the approximations

at the expressions

In this limit, the lines \(\tau _{\Phi } = \tau _{\Phi }(\sigma )\) of constant phase \(\Phi \) satisfy an interesting symmetry relation. Indeed, for a given value \(\tau \) the lines of constant phase corresponding to \(\Phi \) and \(\Phi +\pi \) are separated in their values of \(\sigma \) and \(\sigma ^{\prime }\) by 4. This property follows from Eq. ( 237 ) which for \(\Phi +\pi \) reads

and a comparison with Eq. ( 237 ) immediately yields the connection

and thus the identity

Next, we derive an explicit expression for \(\tau _{\Phi }\) . Unfortunately, Eq. ( 237 ) is a transcendental equation, that is, the Lambert equation, and cannot be solved directly. However, in the limit of large \(\tau \) we obtain an approximate expression for \(\tau _{\Phi }\) by iteration.

For this purpose, we first cast Eq. ( 237 ) into the form

and replace \(\tau _{\Phi }\) in the slowly varying logarithm by the right-hand side leading us to the formula

We conclude by noting that this expression obviously also satisfies the periodicity property, Eq. ( 240 ).

1.2.2 At the Right Edge of the Complex Plane

Next we consider the case \(0\leq \tau \ll \sigma \) and arrive with

In this limit \(\Sigma \) , and thus \(|\xi |\) grows exponentially with \(\sigma \ln (\sigma /2\pi )\) for increasing \(\sigma \) , but decays like a Gaussian due to the term \(-\tau ^2/(4\sigma )\) for \(\tau \) increasing from zero. Moreover, \(\Phi \) increases linearly with \(\tau \) with a rate mainly given by \(\ln (\sigma /2\pi )\) .

According to Eq. ( 247 ), in this asymptotic limit the lines \(\tau _{\Phi } = \tau _{\Phi }(\sigma )\) of constant phase \(\Phi \) read

and decay with a rate, that is, inversely proportional to \(\ln (\sigma /(2\pi ))\) . In particular, \(\tau = 0\) , that is, the real axis, corresponds to the phase \(\Phi = 0\) , and all phase lines approach it as \(\sigma \rightarrow \infty \) .

In the last step of Eq. ( 248 ), we have neglected the correction term \(3/\sigma \) since in obtaining the general expression, Eq. ( 226 ), of \(\alpha _{\xi }\) we have already neglected terms of the order \(1/s\) .

1.3 Cauchy-Riemann Differential Equations

Next we probe the consistency of our approximations, Eqs. ( 236 ) and ( 246 ), as well as Eqs. ( 237 ) and ( 247 ) for the amplitude \(\Sigma \) as well as the phase \(\Phi \) of the exponential representation

of \(\xi \) by applying the Cauchy-Riemann differential equations

discussed in the Chapter “Insights into Complex Functions” of this volume.

We start with the asymptotic expressions

obtained in the previous section for \(1<\sigma \ll \tau \) .

Indeed, by direct differentiation we find

in complete agreement with the Cauchy-Riemann differential equation, Eq. ( 250 ).

Moreover, we also obtain the result

which is consistent with the expression

that is, with Eq. ( 256 ) when we neglect the terms proportional to \(1/\tau \) . Indeed, in the derivation of the expression, Eq. ( 253 ) for \(\Phi \) we have already neglected contributions of this order.

Next, we address the approximations

valid for \(0\leq \tau \ll \sigma \) , and find immediately

where in the last step we have neglected the term \((\tau /\sigma )^2\ll 1\) . Hence, we have verified the Cauchy-Riemann differential equation, Eq. ( 250 ).

Moreover, we obtain from Eq. ( 258 ) the relation

and Eq. ( 259 ) yields

Hence, we satisfy the Cauchy-Riemann differential equation, Eq. ( 251 ), in the approximation \(\sigma \ll 1\) .

Appendix 5: Special Examples of Normalized Gauss Sums

In Sect. 5 we have shown that the normalized Gauss sum

is given by a phase factor whose phase \(\beta \) is determined by the Dirichlet character \(\chi \) . We now illustrate this result by evaluating the normalized Gauss sum of three different characters.

For this purpose, we first consider the normalized Gauss sums associated with the Dirichlet characters forming the Titchmarsh counterexample. In these examples, the phase is non-vanishing. We then calculate the normalized Gauss sum of a character where the phase does vanish.

1.1 Non-vanishing Phase

We start by evaluating the Gauss sums \(G_1\) and \(G_2\) corresponding to the characters \(\chi _1\) and \(\chi _2\) defined \(\mathrm {mod}\;5\) with the values

and demonstrate that they are phase factors, in complete accordance with Eq. ( 143 ). Since \(\chi _1\) and \(\chi _2\) are the complex conjugate of each other, the corresponding phases must satisfy the symmetry relation, Eq. ( 144 ).

We start by noting that in both characters \(q=5\) . Thus, we find with \(\chi _j(-1) = \chi _j(4-5) = \chi _j(4)\) for \(j=1,2\) , and the definitions Eqs. ( 264 ) and ( 265 ) of \(\chi _1\) and \(\chi _2\) , the result \(\chi _1(-1)=\chi _2(-1)=-1\) which yields with Eq. ( 127 ) the parameter

Hence, the Gauss sum \(G_1\) , corresponding to the character \(\chi _1\) reads

and takes with the help of the definition, Eq. ( 264 ), of \(\chi _1\) , the form

where in the last step we have taken advantage of the \(2\pi \) -periodicity of the Fourier factors.

In terms of trigonometric functions, we find

and the values

finally yield the expression

for the Gauss sum \(G_1\) as a phase factor where

We conclude by briefly addressing the Gauss sum

associated with

Indeed, this symmetry enforces that in \(G_2\) the second and the third term in Eq. ( 267 ) change their signs, but everything else remains the same.

As a result, we arrive at a negative rather than a positive phase, that is,

We emphasize that the sign change in the phase due to the transition from \(G_1\) to \(G_2,\) Eqs. ( 269 ) and ( 272 ), can also be viewed as a consequence of the connection, Eq. ( 271 ), between \(\chi _1\) and \(\chi _2\) , and the symmetry relation, Eq. ( 144 ).

1.2 Vanishing Phase

We conclude by discussing an example of a character \(\chi \) where the phase of the corresponding normalized Gauss sum vanishes. For this purpose, we consider the real-valued character

with an integer k .

Since \(\chi \) is \(\mathrm {mod} \; 4\) we find \(q=4\) and with \(\chi (-1) = \chi (3-4)=\chi (3) = -1\) following from the definition, Eq. ( 273 ), of \(\chi \) , we obtain with the help of Eq. ( 127 ) the value \(\kappa =1.\)

Hence, the normalized Gauss sum, Eq. ( 263 ), takes the form

which with the definition, Eq. ( 273 ), of the character \(\chi \) reads

For the character \(\chi \) , given by Eq. ( 273 ), we obtain indeed a vanishing phase \(\beta \) for G .

Appendix 6: Functional Equation of Generalized Jacobi Theta Function

Since the functional equation

of the generalized Jacobi theta function plays a crucial role in the analytic continuation of the Dirichlet L-functions, as discussed in Sect. 6 , we devote the present Appendix to a brief summary of the derivation. This analysis also brings out most clearly the origins of the Gauss sum G and the character in its complex conjugate form.

Here we proceed in two steps: (i) We first show that due to the appearance of the character \(\chi \) and the term \(n^\kappa \) in the definition of \(\omega \) we can express the summation in \(\omega \) from unity to infinity, into one from minus infinity to plus infinity by merely introducing a factor \(1/2\) . This additional symmetry is the deeper reason for the simplicity of the functional equation, Eq. ( 275 ), of the Jacobi theta function for \(\Lambda ,\) compared to the one for \(\xi \) . (ii) Next we take advantage of the periodicity of the character together with the Poisson summation formula, and arrive at the functional equation, Eq. ( 275 ), of \(\omega \) corresponding to \(\Lambda .\)

We conclude by briefly comparing the derivation of the functional equation of the Jacobi theta function \(\omega \) corresponding to \(\Lambda \) to that for \(\xi \) . In particular, we show that the origin of the quadratic polynomial \(s(s-1)\) and the off-set \(1/2\) in the analytic continuation, Eq. ( 43 ), of \(\xi \) is the fact that the term \(n=0\) in \(\omega \) corresponding to \(\xi \) is nonzero.

1.1 Extension of Summation

We start by verifying the representation

For this purpose we consider the sum

and decompose S into a sum over all positive, and one over all negative integers including zero, that is,

where in the sum over negative n we have replaced \(-n\) by \(n.\)

When we recall the connection, Eq. ( 128 ), between \(\chi (-n)\) and \(\chi (n)\) together with \(\chi (0) = 0\) , Eq. ( 119 ), we find the expression

The values, Eq.( 127 ), of \(\kappa \) together with the relation \(\chi (-1) = \pm 1\) yield the identity

and the definition, Eq. ( 277 ), of \(\omega \) the desired representation, Eq. ( 276 ).

1.2 Emergence of Gauss Sum

Since the character \(\chi \) satisfies the periodicity condition, Eq. ( 116 ), it is useful to apply the summation formula

valid for any appropriately converging sum of coefficients \(d(n)\) , to S in the form

where we have introduced the abbreviation

Indeed, with the help of Eq. ( 116 ) we immediately find the representation

of S which with the Poisson summation formula

takes the form

Finally, the new integration variable

yields the expression

where we have interchanged the two summations, and have recalled the definition, Eq. ( 130 ), of the Gauss sum \(\tilde {G}(\chi ,m).\)

Next we use the reduction formula, Eq. ( 129 ), which allows us to factor \(\tilde {G}(\chi ,1)\) out of the sum over m and to find

With the definition of h , Eq. ( 282 ), and the integral formula

we finally arrive at

that is, the functional equation, Eq. ( 275 ), of \(\omega \) when we use the relation, Eq. ( 276 ). Moreover, we have recalled the definition, Eq. ( 142 ), of the Gauss sum.

1.3 Comparison to Jacobi Theta Function of \(\xi \)

We conclude by briefly comparing and contrasting this derivation of the functional equation of the Jacobi theta function of \(\Lambda \) to the corresponding one for the Jacobi theta function of \(\xi \) , defined by Eq. ( 21 ) and discussed in Appendix 1.

In contrast to the generalized Jacobi theta function, Eq. ( 154 ), where the term \(n=0\) vanishes due to \(\chi (0)=0\) , Eq. ( 119 ), the contribution in the one for \(\xi \) is unity. This difference is important since the term in Eq. ( 202 ) independent of \(\omega \) is a consequence of the subtraction of the term \(n=0\) in Eq. ( 195 ). As a result, in the corresponding functional equation, Eq. ( 275 ), for the generalized Jacobi theta function only the contribution proportional to \(\omega (1/x)\) enters.

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Tschaffon, M.E.N., Tkáčová, I., Maier, H., Schleich, W.P. (2023). A Primer on the Riemann Hypothesis. In: Citro, R., Lewenstein, M., Rubio, A., Schleich, W.P., Wells, J.D., Zank, G.P. (eds) Sketches of Physics. Lecture Notes in Physics, vol 1000. Springer, Cham. https://doi.org/10.1007/978-3-031-32469-7_7

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Science News

Mathematicians report possible progress on proving the riemann hypothesis.

A new study of Jensen polynomials revives an old approach

STILL ELUSIVE   Researchers may have edged closer to a proof of the Riemann hypothesis — a statement about the Riemann zeta function, plotted here — which could help mathematicians understand the quirks of prime numbers.

Jan Homann/Wikimedia Commons

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By Emily Conover

May 24, 2019 at 12:03 pm

Researchers have made what might be new headway toward a proof of the Riemann hypothesis, one of the most impenetrable problems in mathematics. The hypothesis, proposed 160 years ago, could help unravel the mysteries of prime numbers.

Mathematicians made the advance by tackling a related question about a group of expressions known as Jensen polynomials, they report May 21 in Proceedings of the National Academy of Sciences . But the conjecture is so difficult to verify that even this progress is not necessarily a sign that a solution is near ( SN Online: 9/25/18 ).

At the heart of the Riemann hypothesis is an enigmatic mathematical entity known as the Riemann zeta function. It’s intimately connected to prime numbers — whole numbers that can’t be formed by multiplying two smaller numbers — and how they are distributed along the number line. The Riemann hypothesis suggests that the function’s value equals zero only at points that fall on a single line when the function is graphed, with the exception of certain obvious points. But, as the function has infinitely many of these “zeros,” this is not easy to confirm. The puzzle is considered so important and so difficult that there is a $1 million prize for a solution , offered up by the Clay Mathematics Institute.

But Jensen polynomials might be a key to unlocking the Riemann hypothesis. Mathematicians have previously shown that the Riemann hypothesis is true if all the Jensen polynomials associated with the Riemann zeta function have only zeros that are real, meaning the values for which the polynomial equals zero are not imaginary numbers — they don’t involve the square root of negative 1. But there are infinitely many of these Jensen polynomials.

Studying Jensen polynomials is one of a variety of strategies for attacking the Riemann hypothesis. The idea is more than 90 years old, and previous studies have proved that a small subset of the Jensen polynomials have real roots. But progress was slow, and efforts had stalled.

Now, mathematician Ken Ono and colleagues have shown that many of these polynomials indeed have real roots, satisfying a large chunk of what’s needed to prove the Riemann hypothesis.

“Any progress in any direction related to the Riemann hypothesis is fascinating,” says mathematician Dimitar Dimitrov of the State University of São Paulo. Dimitrov thought “it would be impossible that anyone will make any progress in this direction,” he says, “but they did.”

It’s hard to say whether this progress could eventually lead to a proof. “I am very reluctant to predict anything,” says mathematician George Andrews of Penn State, who was not involved with the study. Many strides have been made on the Riemann hypothesis in the past, but each advance has fallen short. However, with other major mathematical problems that were solved in recent decades, such as Fermat’s last theorem ( SN: 11/5/94, p. 295 ), it wasn’t clear that the solution was imminent until it was in hand. “You never know when something is going to break.”

The result supports the prevailing viewpoint among mathematicians that the Riemann hypothesis is correct. “We’ve made a lot of progress that offers new evidence that the Riemann hypothesis should be true,” says Ono, of Emory University in Atlanta.

If the Riemann hypothesis is ultimately proved correct, it would not only illuminate the prime numbers, but would also immediately confirm many mathematical ideas that have been shown to be correct assuming the Riemann hypothesis is true.

In addition to its Riemann hypothesis implications, the new result also unveils some details of what’s known as the partition function , which counts the number of possible ways to create a number from the sum of positive whole numbers ( SN: 6/17/00, p. 396 ). For example, the number 4 can be made in five different ways: 3+1, 2+2, 2+1+1, 1+1+1+1, or just the number 4 itself.

The result confirms an earlier proposition about the details of how that partition function grows with larger numbers. “That was an open question … for a long time,” Andrews says. The real prize would be proving the Riemann hypothesis, he notes. That will have to wait.

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Things Of Interest

The riemann hypothesis for dummies.

The Riemann Hypothesis is a problem in mathematics which is currently unsolved.

To explain it to you I will have to lay some groundwork.

First: complex numbers , explained. You may have heard the question asked, "what is the square root of minus one?" Well, maths has an answer and we call it i . i multiplied by i equals -1. If the real number line ...-4, -3, -2, -1, 0, 1, 2, 3, 4... is represented as a horizontal line, then the numbers ...-4i, -3i, -2i, -i, 0, i, 2i, 3i, 4i... can be thought of as the vertical axis on this diagram. The whole plane taken together is then called the complex plane . This is a two-dimensional set of numbers.

Every complex number can be represented in the form a + b i. For real numbers, we simply take b =0.

Next: functions. In mathematics, a function is a black box which, when you put a number into it, spits a different number out. A function is represented by a letter - usually " f ". If you put a number x into the function you call f , then what f then spits out is written " f ( x )".

In most cases there is a convenient way to express f ( x ) in terms of x . For example, f ( x )= x 2 is a very simple function. Whatever x you put in, you'll get x 2 out. f (1)=1. f (2)=4. f (3)=9. And so on.

You're probably most familiar with real functions , or functions where you put a real number in and always get a real number out. HOWEVER. There's nothing stopping you from putting these weird new complex numbers into a function. For example, if f ( x )= x 2 and we let x =i, which is the square root of minus one I mentioned above, then you'll get f (i)=-1. That's just the beginning of what's more generally known as complex functions - where you can put any complex number a + b i in and get (potentially) any complex number out.

The Riemann Zeta Function is just such a complex function. "Zeta" is a Greek letter which is written "ζ". For any complex number a + b i, ζ ( a + b i) will be another complex number, c + d i.

The actual description of the Zeta Function is too boringly complicated to explain here.

Now, a zero of a function is (pretty obviously) a point a + b i where f ( a + b i)=0. If f ( x )= x 2 then the only zero is obviously at 0, where f (0)=0. For the Riemann Zeta Function this is more complicated. It basically has two types of zeros: the "trivial" zeroes, that occur at all negative even integers, that is, -2, -4, -6, -8... and the "nontrivial" zeroes, which are all the OTHER ones.

As far as we know, all the nontrivial zeroes occur at 1/2 + b i for some b . No others have been found in a lot of looking... but are they ALL like that? The Riemann Hypothesis suggests that they are... but nobody has yet been able to prove it.