The magic of Mangoldt summoning Riemann’s miraculous miniscules-the nontrivial zeros.
In response to observations initiated by Matt McIrvin of a sum of exponentials of the imaginary part of the non-trivial zeroes of the Riemann zeta function, assuming the Riemann hypothesis is true, as presented on a stream through Mathstackexchange (MSE), Mathoverflow (MO), and the n-Category Cafe. One thread is the MO-Q Quasicrystals and the Riemann Hypothesis posed by John Baez.
The main actors are the Riemann zeta function , the Landau Xi function (aka, the Riemann Xi function with the two poles removed), the von Mangoldt function , the Chebyshev function (aka, the von Mangoldt summatory function), and the Riemann jump function (aka, the Riemann prime number counting function) with Mellin, Heaviside, and Dirac directing, with a cameo by Fourier.
We formally rederive a relationship beween the zeroes of the Riemann zeta function and the powers of the primes (PP henceforth will be our acronym for powers of the primes or prime powers, the , where is a prime and ) that manifests itself in spikes observed at locations of the imaginary part of the nontrivial zeroes–Dirac delta functions arising from taking a Fourier transform of the derivative of a function that sums over the log of PP, the Mangoldt summatory function, aka, the Chebyshev staircase function, aka, the morphed Riemann jump function for counting the primes.
In fact, the Riemann jump function and the Chebyshev staircase function are two sides of the same coin. The Riemann is a sum of Heaviside step functions, , with the edges of the steps located at the powers of the primes (PP) while the Chebyshev gives the same but with edges located at the log of the PP with different step lifts. One can simply be rewritten into the other with a change of parameters. Consequently, taking the derivative of the functions gives us Dirac delta functions located at PP or, alternatively, their logs. And both are directly related to an inverse Mellin transform of the logarithmic derivative of the Riemann zeta.
Basic Algorithm: Logging product formulas for polynomials
Step I: Factor the polynomial or rational function,
Step II: Take the logarithmic derivative,
Step III: Take the inverse Mellin transform:
for , i.e., putting the line of integration to the left of all the zeros, and closing the contour counter-clockwise to the right,
Step IV: Substitute for for obtaining,
Step V: Extend the result as an even function to ,
Step VI: Take the Fourier transform to obtain Dirac delta functions at the zeros and the pole of and their negatives,
The basic algorithm is to take logarithmic derivatives of product formulas for a “polynomial” to get a sum of simple poles at the zeroes , or at some other parameters characterizing the “polynomial”, and then applying the linear inverse Mellin transform to turn these poles into a sum of exponentiated terms . Making a simple change of variable gives us a sum of exponentials of the zeroes , or the other parameters–tantamount to taking an inverse Laplace transform of the simple poles.
Our “polynomials” are the Riemann zeta, expressed by the Euler product formula for zeta in terms of the PP , and its equivalent the Landau Xi function, expressed by the Hadamard product formula in terms of its nontrivial zeros, which are the same as zeta’s. The Landau Xi function is the Riemann zeta function with its pole and trivial zeroes removed by multiplying by some fairly simple factors–the most complicated being a gamma function whose singularities remove the simple zeroes (remove them initially but reintroduce them in the subsequent analysis through the poles of the digamma function–the logarithmic derivative of the gamma function.)
In the final analysis we have a relation among the locations of PP through the derivative of the Riemann jump function; the locations of the log of the powers of the primes through the derivative of the Chebyshev function, both derivations related to the inverse Mellin transform of the logarithmic derivative of Euler’s product formula; and the locations of the non-trivial zeros of the Riemann zeta function through the inverse Mellin transform of the logarithmic derivative of Hadamard’s product formula for the Landau Xi, in terms of the nontrivial zeros.
Detailed Formal Analysis
Let’s work through the analysis in more detail.
With the Heaviside step function and designating a prime,
where are the natural numbers, since
for with a natural number and vanishes otherwise. Then employing the Dirac delta function
Now look at a property of the Mellin transform. If
then for suitably behaved functions
and, from the earlier post on the Riemann jump function,
implying, for ,
Now make the change of variable , giving
which agrees with the rep above of the zeta fct. ratio and the inverse Laplace transform rep of a delta fct.
The last linear contour integral is essentially an inverse Laplace transform. At this point, it would be good to do a sanity check by numerically evaluating the line integral over finite limits by replacing the infinities by , some finite extent. This will give sinc functions for the delta functions. But, let’s continue.
To get some relation to the non-trivial zeros, multiply the top and bottom of the zeta function ratio to convert the denominator to Landau’s , an entire function symmetric about , with no trivial zeros, and the same non–trivial zeros as the Riemann zeta :
Let’s see if we can tease out a by taking the derivative of and comparing it to the numerator of our ratio. We can then make use of the Hadamard product representation of this log to relate this to pairs of the non-trivial zeros.
Differentiating term by term and using the digamma or psi function (due to a conflict of notation with the Chebyshev function, I shall use for the digamma function):
and our ratio becomes
The Hadamard product formula gives
where the sum is over the zeros above the real axis and the lower zeroes are entered through taking the complex conjugate.
Taking the derivative,
Now appealing to Mangoldt’s formula, formally derived in Appendix II from the formula’s above, we have for
Taking the derivative and relating this to our other expression above,
Multiplying both sides by , and suppressing the Heaviside step function, gives
and, letting for ,
We can extend this equation to as an even function on both sides by changing to on both sides and averaging the two equations together. Taking the Fourier transform w.r.t. will give an odd function of Dirac delta functions on the LHS located at absiccas equal to the imaginary part of the nontrivial zeros.
The 2007 pdf “What is the Riemann hypothesis?” by Mazur and Stein contains plots of partial sums of the and a discussion. I assume their book extends and elaborates on these fantastic facets of the Riemann zeta.
Appendix I: Mellin Transforms and Inverse
and for suitably chosen .
1) for so that the upper evaluation vanishes.
for if we truncate the vertical integration line with the infinities replaced by some positive finite and close the contour counter-clockwise to the left with a semicircle of radius for , the closed contour contains no singularities, so the contour integral evaluates to and the integral along the semicircle tends to zero as tends to infinity, so the integration over the vertical line evaluates to zero. On the other hand, if we close clockwise to the right with a semicircle, which introduces an overall negative sign, with , the closed contour contains a simple pole at the origin and evaluates to unity while the integral along the semicircle vanishes in the limit as tends to positive infinity.
2) for so that the lower eval vanishes.
3) for .
for , and for ,
5 a.) Now to tackle the inverse Mellin transform of the digamma function using the representation
for (or 0 for the integral rep?), where is the Euler-Masheroni constant.
for , but analytically continues to all complex numbers except the negative natural numbers .
5 b) ,
so the inverse Mellin transform for closing clockwise to the right for gives
Note the Mellin transform of this expression would have to be regularized to obtain the digamma expression again. This is a common occurence. In fact, regularization is applied to get a Mellin transform for the continuation of the gamma function to the left of its singularities, and the Euler integral expression for the digamma itself is a differently regularized Mellin transform.
5 c) For Mangoldt’s explicit formula, we need to evaluate a slightly different inverse transform of a digamma function:
for closing clockwise to the right for gives
6) A general Mellin transform relation between the Mellin transform of a function and that of an integral of the function through integration by parts:
so integrating over from to and rearranging terms,
where for suitably chosen .
Alternatively, with ,
7) For example, for ,
This is consistent with
for and ,
8) To evaluate double poles:
for and suitably decaying at infinity to the right of with no other poles or branch cuts in that region. Depending on , the evaluation could be extended to .
(At first I thought I needed to eval double poles, but I found a way to circumvent it. I leave this for illustrative purposes of the properties of the inverse Mellin transform.)
(We could also collapse our integration line to just beneath and above the real axis, like a Hankel contour, to the right of and generate an integral along the real axis containing the derivative of a Dirac delta function.)
Appendix II: Mangoldt’s explicit formala for the Chebyshev function
To relate the analysis here to other derivations, note that , so
and , and, in particular,
under the obvious change of variable from to .
In the main body of this post, we find the relation between Chebyshev’s function, or Mangoldt’s summatory function, and an inverse Mellin transform of the log derivative of the Riemann zeta. Evaluating this transform, and equating it to other expressions above we arrive at Mangoldt’s explicit formula.
Evaluating term by term and using the identity for the digamma in Appendix I Example 5c,
Since symmetry gives , and according to “Relations and positivity results for the derivatives of the Riemann function” by Coffey
Appendix III: Dirac delta combs and approximations
Consider the nascent Dirac delta function given by
Then the sifting property holds on functions continuous around the origin and suitably behaved elsewhere:
This can be shown for suitable functions by using the Fourier convolution theorem and then taking the limit, but works for a wider class of functions also.
Note that taking the Fourier transform over finite limits gives us our oscillating function, a nascent delta function, rather than a sharp spike.
Now consider a sum of exponentials of uniformly spaced purely real numbers symmetric about the origin when the spacing between the numbers is so that for and . Then
for any integer , so the sum of exponentials gives a periodic function which behaves about as
which tends to
as and tend to infinity with a fixed constant, giving us our Dirac comb
Of course, the zeros of the Riemann zeta are not evenly spaced, so we can’t expect to find a Dirac comb in our case, but the analysis does suggest at best summing over a finite number of zeros will give us oscillating sinc functions rather than Dirac delta functions.
Appendix IV: Riemann’s explicit formula (added Oct. 2, 2019)
where , the logarithmic integral, and, as above, denotes the nontrivial zeros.
From the main text above, we have
so for , we have from the analysis in the main text
and from differentiating Riemann’s explicit formula, we obtain consistently
Appendix V: More basic Mellin transform properties
Dirichlet series are best thought of as inhabiting the inverse Mellin transform space and as the dual of a Dirac delta distribution in real space as shown in my earlier post on the Riemann jump function:
Note this becomes if or, equivalently, . See the famous example in my answer to the Math Stackexchange question “Does the functional equation have any nontrivial solutions … ?”
What is the inverse Mellin transform of for ?
Appendix VI: Some useful properties of the Dirac delta
1) Sifting property
We define the Dirac delta “function” by its sifting property acting within an integral on a function suitably behaved about a small neighborhood of the point where the argument of the delta function vanishes:
so we have
The singularity at the origin is no problem if since by integration by parts
This has no singularity at the origin if doesn’t.
Now taking the derivative of the sift equation above, we obtain
(In fact, the relevant functions dealt with in the main text are null at the origin and for some interval to the right of it.)
Changing variables we derive other properties.
2) Even symmetry property
Let , then
, but also
so the Dirac delta is an even “function”; i.e.,
3) Reciprocal scaling property
let . Then for ,
and for ,
4) Composition property
Let with only one zero and that at , then since we are only concerned with where the argument of the delta vanishes
5) The derivative property
Clearly the derivative of the delta is odd since the delta is even:
Use the limit of the Newton-Fermat quotient,
in the limit gives
Similarly, by a shift, or change of varisble,
6) The BYOYB property
Beware! The magical delta can bite you on your butt at a moment’s notice if you aren’t careful. For example, the nascent Dirac delta function discussed above
is sometimes said to approach the Dirac delta as approaches infinity yet it lacks the important symmetry
that follows from the properties above. This apparent quandary is resolved once the the nascent function is embedded in an integral:
Trying to corroborate my analysis, I found consistent results and extensions
1) “Notes on the Riemann hypothesis” by Peraz-Marco
2) The Prime Number Theorem: a proof outline: at the Number Theory and Physics Archives
3) Chebyshev function at Wikipedia
4) Beurling zeta functions, generalized primes, and fractal membranes by Hilberdink and Lapidus
5) Spectral analysis and the Riemann hypothesis by Lachaud
6) Notes on the Riemann hypothesis by Perez-Marco
7) An essay on the Riemann Hypothesis by Alain Connes
Added on Oct. 3, 2019:
8) “The Riemann hypothesis explained” by Veisdal (blog post). A quick general intro to the Riemann zeta, RH, and the prime number theorem.
9) “Riemann’s Explicit Formula” by Sean Li. Derivations with convergence arguments provided.
10) “A history of the prime number theorem” by Anita Alexander
11) “A history of the prime number theorem” by Goldstein
12) “Mellin convolution and its extensions, Perron formula, and explicit formulae” by Jose Javier Garcia Moreta. An analysis also based on simple Mellin transform properties.
13) Convergence of Riemann spectrum/Fourier transform of prime powers a MSE question posed by Joe Knapp and answered by reuns
14) “Twenty female mathematicians” by H. Williams. (See the section on Mirzakhani.)
15) “Some remarks on the Riemann zeta distribution” by Allan Gut