Salem's equivalence of the Riemann hypothesis asserts that the hypothesis is true if and only if $ g(t) = 0 $ is the only nontrivial bounded measurable solution (for fixed $ 1/2 < r < 1 $) of the following integral equation:
$ \int_{0}^{\infty} \frac{t^{r-1} g(t)}{e^{xt}+1}\,dt = 0, \qquad x>0. $
Recent works have investigated this integral equation and shown that various classes of bounded measurable functions, subject to suitable growth assumptions, cannot furnish counterexamples to Salem's criterion. These approaches typically employ Mellin inversion, Widder–Lambert–type transforms, or distribution-theoretic methods. The aim of this paper is to study the associated nonhomogeneous Salem-type equation
$ f(x) = g(x) + \lambda \int_0^\infty \frac{t^{\,r-1} f(t)}{1+e^{xt}}\,dt, \qquad x>0, $
where $ g $ is a given measurable function, $ r\in(1/2, 1) $, and $ \lambda\in\mathbb{R} $, from the viewpoint of fixed–point theory. We show that the corresponding integral operator does not map the space of bounded measurable functions into itself on $ (0, \infty) $. We therefore introduce a suitable weighted complete function space in order to apply Banach's contraction principle. The nonhomogeneous equation is important because it is a more general form of Salem's equation. As we shall see, its analysis allows one to identify precisely why Banach's contraction principle succeeds for the nonhomogeneous equation in an appropriate weighted space, yet does not directly settle Salem's original integral equation. This paper suggests that possible future directions may include the use of generalized metric spaces and generalized contraction mappings to study Salem's integral equation.
Citation: Irshad Ayoob, Nabil Mlaiki. A Banach fixed–point approach to Salem's nonhomogeneous integral equation related to the Riemann hypothesis[J]. AIMS Mathematics, 2026, 11(4): 12155-12177. doi: 10.3934/math.2026499
Salem's equivalence of the Riemann hypothesis asserts that the hypothesis is true if and only if $ g(t) = 0 $ is the only nontrivial bounded measurable solution (for fixed $ 1/2 < r < 1 $) of the following integral equation:
$ \int_{0}^{\infty} \frac{t^{r-1} g(t)}{e^{xt}+1}\,dt = 0, \qquad x>0. $
Recent works have investigated this integral equation and shown that various classes of bounded measurable functions, subject to suitable growth assumptions, cannot furnish counterexamples to Salem's criterion. These approaches typically employ Mellin inversion, Widder–Lambert–type transforms, or distribution-theoretic methods. The aim of this paper is to study the associated nonhomogeneous Salem-type equation
$ f(x) = g(x) + \lambda \int_0^\infty \frac{t^{\,r-1} f(t)}{1+e^{xt}}\,dt, \qquad x>0, $
where $ g $ is a given measurable function, $ r\in(1/2, 1) $, and $ \lambda\in\mathbb{R} $, from the viewpoint of fixed–point theory. We show that the corresponding integral operator does not map the space of bounded measurable functions into itself on $ (0, \infty) $. We therefore introduce a suitable weighted complete function space in order to apply Banach's contraction principle. The nonhomogeneous equation is important because it is a more general form of Salem's equation. As we shall see, its analysis allows one to identify precisely why Banach's contraction principle succeeds for the nonhomogeneous equation in an appropriate weighted space, yet does not directly settle Salem's original integral equation. This paper suggests that possible future directions may include the use of generalized metric spaces and generalized contraction mappings to study Salem's integral equation.
| [1] | P. Borwein, S. Choi, B. Rooney, A. Weirathmueller, The Riemann hypothesis: A resource for the afficionado and virtuoso alike, Springer, 2008. https://doi.org/10.1007/978-0-387-72126-2 |
| [2] | K. Broughan, Equivalents of the Riemann Hypothesis, Volume two: Analytic equivalents, Cambridge: Cambridge University Press, 2017. https://doi.org/10.1017/97811081/78266 |
| [3] | K. Broughan, Equivalents of the Riemann hypothesis, Volume 1: Arithmetic equivalents, Cambridge: Cambridge University Press, 2017. https://doi.org/10.1017/9781108178228 |
| [4] | F. T. Wang, A note on the Riemann zeta-function, Bull. Amer. Math. Soc., 52 (1946), 319–321. |
| [5] |
N. Levinson, On closure problems and zeros of the Riemann zeta function, Proc. Amer. Math. Soc., 7 (1956), 838–845. https://doi.org/10.1090/S0002-9939-1956-0081922-4 doi: 10.1090/S0002-9939-1956-0081922-4
|
| [6] |
V. V. Volchkov, On an equality equivalent to the Riemann hypothesis, Ukr. Math. J., 47 (1995), 491–493. https://doi.org/10.1007/BF01056314 doi: 10.1007/BF01056314
|
| [7] | R. Salem, Sur une proposition équivalente à l'hypothèse de Riemann, C. R. Acad. Sci. Paris, 236 (1953), 127–128. |
| [8] |
M. Balazard, E. Saias, M. Yor, Notes sur la fonction $\zeta$ de Riemann, 2, Adv. Math., 143 (1999), 284–287. https://doi.org/10.1006/aima.1998.1797 doi: 10.1006/aima.1998.1797
|
| [9] |
S. K. Sekatskii, S. Beltraminelli, D. Merlini, On equalities involving integrals of the logarithm of the Riemann $\zeta$-function and equivalent to the Riemann hypothesis, Ukr. Math. J., 64 (2012), 247–261. https://doi.org/10.1007/s11253-012-0642-0 doi: 10.1007/s11253-012-0642-0
|
| [10] |
S. Yakubovich, Integral and series transformations via Ramanujan's identities and Salem's type equivalences to the Riemann hypothesis, Integr. Transf. Spec. F., 25 (2014), 255–271. https://doi.org/10.1080/10652469.2013.838762 doi: 10.1080/10652469.2013.838762
|
| [11] |
B. J. González, E. R. Negrín, Inversion formulae for a Lambert-type transform and the Salem's equivalence to the Riemann hypothesis, Integr. Transf. Spec. F., 34 (2023), 614–618. https://doi.org/10.1080/10652469.2023.2169284 doi: 10.1080/10652469.2023.2169284
|
| [12] |
B. J. González, E. R. Negrín, Approaching the Riemann hypothesis using Salem's equivalence and inversion formulae of a Widder–Lambert-type transform, Integr. Transf. Spec. F., 35 (2024), 291–297. https://doi.org/10.1080/10652469.2024.2321377 doi: 10.1080/10652469.2024.2321377
|
| [13] |
E. R. Negrín, J. Maan, New inversion formulae for the Widder–Lambert and Stieltjes–Poisson transforms, Axioms, 14 (2025), 291. https://doi.org/10.3390/axioms14040291 doi: 10.3390/axioms14040291
|
| [14] | S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181. |
| [15] | R. Kannan, Some results on fixed–points, Bull. Calcutta Math. Soc., 10 (1968), 71–76. |
| [16] | S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulg. Sci., 25 (1972), 727–730. |
| [17] | S. Reich, fixed–points of contractive functions, Boll. Unione. Mat. Ital., 4 (1972), 26–42. |
| [18] |
G. Hardy, T. D. Rogers, A generalization of a fixed–point theorem of Reich, Can. Math. Bull., 16 (1973), 201–206. https://doi.org/10.4153/CMB-1973-036-0. doi: 10.4153/CMB-1973-036-0
|
| [19] |
J. Brzdek, N. Eghbali, On approximate solutions of some delayed fractional differential equations, Appl. Math. Lett., 54 (2016), 31–35. https://doi.org/10.1016/j.aml.2015.10.004 doi: 10.1016/j.aml.2015.10.004
|
| [20] | J. Meszáros, A comparison of various definitions of contractive type mappings, Bull. Calcutta Math. Soc., 84 (1992), 167–194. |
| [21] |
B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226 (1977), 257–290. https://doi.org/10.1090/S0002-9947-1977-0433430-4 doi: 10.1090/S0002-9947-1977-0433430-4
|
| [22] | B. E. Rhoades, Contractive definitions revisited, In: Topological methods in nonlinear functional analysis, American Mathematical Society, 1983,189–205. https://doi.org/10.1090/conm/021 |
| [23] | B. E. Rhoades, Contractive definitions, In: Nonlinear analysis, Singapore: World Scientific Publishing, 1987,513–526. |
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Irshad Ayoob, Nabil Mlaiki. A Banach fixed–point approach to Salem's nonhomogeneous integral equation related to the Riemann hypothesis[J]. AIMS Mathematics, 2026, 11(4): 12155-12177. doi: 10.3934/math.2026499
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