Research Article | Open Access
An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution
Two identities extracted from the literature are coupled to obtain an integral equation for Riemann’s function and thus indirectly. The equation has a number of simple properties from which useful derivations flow, the most notable of which relates anywhere in the critical strip to its values on a line anywhere else in the complex plane. From this, both an analytic expression for , everywhere inside the asymptotic critical strip, as well as an approximate solution can be obtained, within the confines of which the Riemann Hypothesis is shown to be true. The approximate solution predicts a simple, but strong correlation between the real and imaginary components of for different values of σ and equal values of t; this is illustrated in a number of figures.
The Riemann zeta function is well known to satisfy a functional equation, and many representations, both integral and series, have been developed over the years for both and its avatar . Additionally, and more significantly, at least two independent contour integral representations are known (, Section (1.4)) and (, equation (7)) either of which could be utilized as the primary definition, from which many of the properties of can be derived. On the contrary, it has been long-ago proven that does not satisfy a differential equation of quite general form ( (and citations therein)); also known are three integral equations which does satisfy: , equation (1.6), , equation (3.29), and , equation (1.5) with . In this work, a series representation of in terms of generalized integro-exponential functions () is coupled with a contour integral representation of these functions, to obtain a new integral equation satisfied by and, equivalently, . This equation has several remarkable properties:(i)Through the action of an integral operator, the value of (the “dependent”) anywhere in the complex plane can be determined with respect to the properties of (the “master”) on a line anywhere else in the punctured complex plane(ii)The transfer function that mediates the aforementioned action is a simple rational polynomial function of and is therefore quite amenable to analysis(iii)The only dependence resides in the transfer function
In this work, after initially listing some definitions and lemmas (Section c), a derivation of the integral equation over a bounded region of the -plane (Section 3) is given, and its analytic continuation over the remainder of the punctured complex plane (Section 4) is obtained. In Section 5, various representations are then used to obtain a few simple integrals involving , corresponding to special values of ; it is also demonstrated that the simple integrals under consideration are convergent.
Following these preliminaries, the general form of the transfer function is carefully presented in a series of Appendices. Based on a particularly useful property of the transfer function presented in Appendix C—it closely mimics a Dirac delta function on the line— a reasonably accurate model for is established, which yields reasonably accurate estimates of the asymptotic nature of inside the critical strip (Section 6). In consequence, it is shown that, asymptoticallyand, within the confines of the model, it is both proven that if , and explained, in transparent terms (Section 7), why this happens (does not happen?). Graphical examples and comparisons are given, demonstrating the accuracy of a predicted universality between the real and imaginary components of at equal values of t but different σ. In Section 8, two well-accepted properties of , notably Universality (, Chapter XI) and Lindelöf’s hypothesis (, Section 9.2) are discussed in the context of the results reported here, and suggestions are made that merit future consideration. Section 9 discusses requirements needed to improve the rigour of the model developed here.
Throughout, I use to denote the independent variable defining . In those instances, where dependence on, for example, t is the focus, I will sometimes just shorten, for example, , for typographical brevity and clarity. Throughout, subscripts ‘R’ and ‘I’ refer to the real and imaginary components of whatever they are attached to. Much use is made of Riemann’s ξ function, defined byas well asboth of which satisfy
2.2. Definitions and Lemmas
Definition. Polar form means that a complex function is written as , where .(i)The following asymptotic limits (, equation (5.6.9); see also , equation (4.12.2)) will be required: and the special case
Lemma 1. Well-known (, equation (4.17.2)) with , this is sometimes referred to as the Riemann–Siegel identity. A simple derivation with follows.
Remark. Apply the identityto (9) to obtain Backlund’s formula which counts the discontinuities (not zeros) of (see , Section 9). This is consistent with Theorem 1 in .(iii)DefineHowever, for brevity, expressions can be abbreviated as follows:
3. LeClair’s Representation
In a recent work, LeClair (, equation (15)) has obtained the following series representation of Riemann’s ξ function:where generically, is the (generalized) “Exponential Integral,” a limiting case of what is elsewhere  referred to as the “Generalized Integro-Exponential Function.” Furthermore, it has been shown (, equation (3)) that the infinite sum in (19) can be readily evaluated as follows:
In his work, LeClair truncated the sum(s) at N terms, referred to the result as an “approximation” and proceeded to obtain approximations to the location of the zeros on the critical line on that basis. Here, the sums are treated as an infinite series representation of , and hence an identity because each series is easily shown to be convergent due to the asymptotic property of (see , equation (2.25)). Similar, but inequivalent series representations will be found in Paris (, equation (1)), Patkowski (, equation (1.20)), and elsewhere. From , some useful integral and contour integral representations of the function are
In (21), is the (upper) incomplete gamma function, and (22) provides the fundamental definition of . The result (23) is given in  (equation (2.6a)). Here, the integration contour, originally defined to enclose the real axis as well as the singularity at in a clockwise direction, has been converted into the line because the integrand vanishes as . This paper investigates the application of (23) to (19).
3.1. A Novel Integral Representation
The following, focuses on the limited range , where , in which case, applying (23) to the first term on the right-hand side of (19) yieldswhere the interchange of integration and summation in (24) is justified if since the sum converges under this condition. The general requirement that imposes the effective constraint for : the contour may be shifted leftwards with impunity since there are no singularities in that direction. Similarly, under the transformation , we find the following expression for the second right-hand side term of (19):again valid for if . In the case that , where , and focussing on , we find the integral equation:which, in terms of , can be rewritten as
Notice that with a simple application of the recursion formula for , the numerator in (28) can be written in terms of (see (3)). Alternatively, under a simple change of variables, with the same condition and the same range of σ, (28) becomeswhere
This result is worthy of a few comments:(i)Its form is almost (exception: see , equation (1.18)) unique among representations of . Usually s-dependence that is formally embedded inside an integral or series representation of appears either as an exponent or buried inside the argument of a transcendental function; here, s-dependence exists only in the form of a coefficient in a simple rational (polynomial) function. This augers well for further analysis and leads to some surprising predictions;(ii)(27) and (28) present a prescription, in the form of an integral transform, for the value of or anywhere in the complex strip , that depends only on its values on a vertical line in the complex plane corresponding to . The region in which (29) is valid is labelled “I” in Figure 1(b) and delineated as everything to the left of the point at in Figure 1(a);(iii)In (29) and (30), the function acts as a transfer function between on the line and elsewhere in the complex plane, through the medium of the integral operator (30). Its properties will be of interest in studying (29) for varying values of the parameters c and s;(iv)Based on (4), it is possible to show (, Theorem 1) that (28) can be cast into a simple closed (Cauchy) contour integral for .
In the following sections, simple choices of s and , and will be applied to (29).
4. Analytic Continuations
In the following, let . By shifting the contour (28) into the region (see Figure 1(a)), various representations are obtained for each of the regions labelled in Figure 1(b), by subtracting the residues of the singularities that are transited in the complex -plane. To enter the region labelled “II” in Figure 1(b), the contour in Figure 1(a) must pass to the right of the fixed pole at , whose residue is given bygiving, for region II,
To cross the green line, either from region II, entering region V, or from region III, entering region IV, the residuecorresponding to the green pole in Figure 1(a) must be subtracted. A similar argument applies for a transition from region V to region IV, the residue (34) must be subtracted. This gives, for region IV,
In particular, this analysis relates the results applicable to region IV to that for region I, including regions where both representations are unconstrained with respect to and equates the result for region V to that for region III. It is emphasized that the regional definitions in Figure 1(b) exclude the lines, thereby defining the punctured regions alluded in the Introduction. Of particular interest, is the value at the point , as well as the values along the horizontal (dotted) line and a vertical line . To reiterate, Figure 1(b) is intended to illustrate the projection of a (complex) space defined by a third axis extending out of the plane of the Figure, corresponding to , onto the (real) plane where .
5. Crossing the Line at c = −3/2
Remark. Several of the identities in this section are (redundantly) declared to be real and marked . This is done as a reminder that unless so-declared, an attempted numerical computer evaluation of that expression will likely fail because the imaginary part of the identity contains a singularity (e.g., (42)).
5.1. The General Case c = −3/2
Disregarding the contour integral representations from which they are obtained, (29), (33), and (35) simply involve several complex functions of two real variables (if ) and ought to be amenable to analysis from that viewpoint. Compare (29) and (33), respectively, below and above the horizontal line demarking regions I and II of Figure 1(b) by setting in (29) and in (33), where . Note that η is a real variable (there is no cut); since none of the terms in either equation (other than ) depend on c, we are free to choose a convenient value for c, subject to the (regional) conditions under which each equation is valid. We have in region I:and in region II
This can be independently verified by a change of variables in both integrals, followed by a series expansion about , leading to
As is often done where discontinuities arise in the theory of functions of a real variable, when it is conventional to assign half the discontinuity at that point . This is equivalent to deforming the contour in (28) to avoid the fixed pole belonging to at by including only half the residue at that point.
Remark. This cannot be done for the other poles since they are located by s (complex), rather than (fixed).
5.1.1. The Case
This result (compared with (, page 204, Lemma β)) can be verified numerically. The need to specify that only the real part of the integrand in (42) is to be used is twofold:(i)The right-hand side is real and so must be the left-hand side(s).(ii)Ostensibly, the integrand of (42) appears to be singular at unless one notes thatwhere γ is Euler’s constant. Therefore, the real part of the integrand converges at and the imaginary part, singular at the origin, integrates to zero by antisymmetry.
5.2. The Case c = −5/4
In regions III and V, writing (35) in full givesor, equivalently
Setting in (45) leads toa result that is valid for all s (see the dotted line in Figure 1(b)). Since the left-hand side is a function of s and the right-hand side is not, it must be true that the derivative of the left-hand side with respect to s vanishes. Performing this calculation (the integral is convergent, so the derivative and integral operator can be interchanged) yields
Consider (47) in the case , which givesand the integrand apparently has a singularity at . However, a simple expansion of the integrand about that point shows thatdemonstrating that the real part of the integrand in (49) is nonsingular at . However, near , it is similarly shown that the imaginary part of the integrand diverges like and the imaginary part of the integral vanishes by antisymmetry about .
In the case , (47) can be written in terms of aswhose integrand appears to become singular at . Writing the integrand in terms of its real and imaginary parts, and noting that is real, we find, for the real partand, for the imaginary part,
Notice that in the former case, there is no t dependence, so in (51), the variable t is in reality a free parameter, consistent with the argument applied to obtain (48). In the latter case, although a singularity exists, the integrand is antisymmetric about , and so the singularities at cancel.
6. An Asymptotic Approximation to
6.1. Integral Equations for the Real and Imaginary Parts of
Consider (37)—region IV—in the general case , written in terms of the transfer function . The case is of interest since the integral will span the 0-line, which connects to the 1-line by reflection (see (8)), and, acting in its capacity as a master function, has many well-known properties (e.g., ), the most relevant one being that it has no complex zeros. In addition, according to Figure 1(b), (37) is valid for all values of σ spanning the critical strip . In this case, the transfer function and its real and imaginary components, shortened, except where necessary, to and , are, respectively,
Furthermore, these functions possess the following symmetry:which symmetry also holds true for all values of c. A complete description of these functions is given in Appendixes A and B. From (37), the basic equation in region IV using is
It is now convenient to rewrite (59) as an integral over the range by first splitting the integral, setting and combining the two halves. The expression obtained is straightforward, although rather lengthy. A second lengthy equation can be obtained by first replacing in (59) and similarly reducing the range to . After splitting into its real and imaginary components, these two equations can be added and subtracted, and with the help of (57) and (58), two new fundamental equations emerge:where
In this manner, because is self-conjugate, the real and imaginary components of are isolated, and expressed in terms of convergent integrals. Furthermore, subtracting (D.2) from (60) yields a simpler form of (60):
For the case , (66) reduces to a relation between on a section of the real line, and on the complex line , specifically
6.2. The Background Terms
Although the functions look formidably complicated when written in full, each can be written in a more transparent form when decomposed using partial fractions. Specifically, with ,
Remark. Although not evident when written in the decomposed form, when expressed in factored form, both (70) and (71) contain an overall factor , a requirement that vanishes when . Furthermore, it is easily seen that , , , and .
As written, the first two terms of each of the above explicitly demonstrate the existence of a pole in the complex -plane at , along with invariance under the symmetry . Since we are in general interested in the case , such poles lie relatively close to the positive real -axis as demonstrated by the nature of the peak(s) in Figure 14 (see Appendix A). By reducing the integral to the range , the influence of poles corresponding to negative values of has been effectively removed from the integrand . Drawing on these observations, it is suggestive that the functions can be separated into two components—pole terms associated with the first two terms of each, plus background terms associated with all four terms, in each of the above; it is further suggestive that only the pole terms will contribute to the integrals (see Figures 16 and 18), thereby allowing an approximate solution to be obtained. How can this be done?
Consider (66). From (68) and (69), it would appear superficially that both and vanish as for large values of t, which immediately demonstrates a potential inconsistency—the left-hand side of (66) vanishes exponentially and so must the right-hand side. Since the remaining integrand factors and contain no t dependence, the only possibility is that dependence must vanish from the (highly oscillatory) integral, and somewhere buried in the higher order asymptotic terms will be found a term with exponentially decreasing t dependence. Furthermore, the possibility also exists that cancellations will occur between the two integrand terms containing and in (66). Any attempt at a numerical evaluation of (66) for reasonably large values of t will immediately demonstrate the truth of this prediction in the form of a severe cancellation of significant digits. An example follows.
Choose and , chosen sufficiently large to distinguish inverse powers of t from exponentially decreasing terms of the form , but not so large that multidigit computer arithmetic must be used. For this case, calculated with 15 digits, we findand the sum is comparable with . Clearly, each of the integrals has an absolute value much greater than the sum of the two, and it is only because of a cancellation of digits that the final result can be found. Here, we see that the cancellation of digits has resulted in the loss of 10 digits from the sum; for larger values of t, the effect will be more significant, and one might despair of utilizing (66) for anything. It will now be shown that these (and other) cancellations can be analysed analytically, and an exponentially decreasing term can be extracted, which is the basis for the approximate model solution.
First of all, for , consider the asymptotic expansion of each of the individual terms composing labelled, respectively, by a second subscript:Clearly, the leading asymptotic dependence cancels between all of “pole” and “background” terms in pairs, demonstrating that the pole terms also contribute significantly to the asymptotic background. This verifies that a numerical evaluation of the integral will be challenging as t increases. More importantly, it is apparent that the overall asymptotic dependence of the part of the integral will have the leading dependence because the multiplicative factor lacks t dependence as noted. But, as also noted, this is inconsistent with (66), where the left-hand side has an exponentially decreasing asymptotic behaviour in t. One explanation for this inconsistency could be that there exists a cancellation between the two terms in the integrand labelled by and to at least order . This would explain the cancellation observed in the numerical results (72) and (73). The following gives the asymptotic expansion of each of the corresponding elements of in analogy to (74)–(77):Again, we see a cancellation between each of the four terms to order as well as a cancellation of the terms of order with the fifth term in (69). And, again we find an inconsistency with (66) because the leading term of is also of order . Define each term corresponding to in the asymptotic series of byand similarly for . When the first two asymptotic terms of the factors in the integrand on the right-hand side of (66) are written in full, we findso thatand clearly there can be no cancellation of terms of equal order in , or between terms containing and . The only remaining possibility is that the integral containing each of these asymptotic forms itself vanishes identically, at least to the orders discussed above. In fact, this must be true to all asymptotic orders in because it is impossible for a series of the form to add up to an exponential of the form . Furthermore, if the integrals vanish as predicted, they must vanish for all σ. Otherwise, (66), which is an exact result cannot be asymptotically and universally true. And somewhere, accompanying all these dominant terms that vanish asymptotically to inverse polynomial order, exist other terms associated with the pole terms, that carry the exponentially vanishing result corresponding to the left hand-side of (66) in the asymptotic limit . Numerically, these lesser terms equate to the sum of (72) and (73). It will now be proven that the integrals do in fact vanish, as predicted, to at least the first few orders in , following which a model will be constructed to extract the terms that instead vanish exponentially.
Based on the first terms of (83) and (84), it is predicted that the sum of the integrals involving must vanish, that is,
Proof. Consider the classical result (, equation (2.15.6)) relating the inverse Mellin transform of to the Jacobi function:valid for . We are interested in the case , so shift the contour by subtracting the residue at and half the residue at resulting in the identityif , after converting the integration limits to and writing as the sum of its real and imaginary parts. By an increasing sequence of higher order derivatives, evaluated at , it is possible to obtain relevant sums using identities given by Romik . The first seven such sums are listed in Appendix D, from which it is possible to obtain the first seven even moments of and odd moments of , listed also in Appendix D. Substituting the moments (D.2)–(D.4) into (89) verifies that the prediction (89) is true. QED.
Employing similar logic, a further prediction arises regarding the next term of order . It too must vanish for all σ. From (83), (84) and (89) we predictNote that σ dependence has disappeared from (92), as expected. By substituting (D.2)–(D.6) into (92) the truth of this prediction can be verified. This sequence of forecasts can be continued to the next recursive level, producing the forecast identity:Again, σ dependence has cancelled of its own volition. The truth of the prediction (93) can similarly be verified by substituting the set of moments (D.2)–(D.8) into (93).
To summarize, we have the following proposition.
Proposition 1 (deduced and partially demonstrated). The above sequence of predictions can be proven in general for all terms on the right-hand side of (66) that asymptotically vanish to inverse polynomial order independently of σ.
Whatever remains must then be of exponentially decreasing order and must therefore equate to the left-hand side of (66), and the challenge is to somehow isolate such term(s). The cancellation of integrals of asymptotically vanishing inverse polynomial order explains the loss of significant digits observed between (72) and (73) and suggests that such loss of significant digits will only increase for larger values of t, as the interested reader can verify for herself.
Expressed another way, the crucial point is that although the peaks shown in Figure 14 originate from the pole terms, these pole terms also contribute to the background terms of inverse polynomial order that cancel. Nonetheless, the signal associated with the pole terms naturally separates from the background. The separation of signal from background is the essence of the approximate solution to be presented here.
To gain further insight into these issues, consider Figure 2(a) which shows (the absolute value of) the integrand:as a function of for several values of t at . Similarly, Figure 2(b) shows the same result for the second term in the integrand defined by
Three properties are immediately evident: most of the absolute value of both integrands occurs near , both and decrease exponentially with increasing and are of similar magnitude—a necessity if they are to cancel numerically (see (72) and (73)). A fourth significant property is that the pole term, embedded in at is clearly visible above the exponentially decreasing background in each curve of Figure 2(a), although it lies many orders of magnitude below the numerically significant portion of the integrand. In fact, it appears to be enhanced as t increases, as might be expected because the magnitude of its peak varies as t—see (C.1)—whereas the background varies as . However, in Figure 2(b) the zero of is not readily apparent because of the resolution of each curve; moreover, each of the curves in Figure 2(b) is consistent with what appears to be the background as discussed previously. It will now be shown that these observations can be used to extract a signal from the background, and in so doing obtain an approximate solution to (66).