To study point groups, their irreducible characters are essential. The table of irreducible characters of the icosahedral group A_5 is usually obtained by using its duality to the dodecahedral group. It seems that there is no literature which gives a routine computational way to complete it. In the works of Harter and Allen, a computational method is given and the character table up to the tetrahedral group A_4 using the group algebra table and linear algebra. In this paper, we employ their method with the aid of computer programming to complete the table. The method is applicable to any other more complicated groups.
@article{kms2023,title={Irreducible characters of the icosahedral group},author={Kanemitsu, S. and Mehta, Jay and Sun, Y.},year={2023},journal={Nanosystems: Phys. Chem. Math.},number={4},volume={14},pages={405-412},}
On Popov’s Explicit Formula and the Davenport Expansion
We shall establish an explicit formula for the Davenport series in terms of trivial zeros of the Riemann zeta-function, where by the Davenport series we mean an infinite series involving a PNT (Prime Number Theorem) related to arithmetic function an with the periodic Bernoulli polynomial weight (Formula presented.) and PNT arithmetic functions include the von Mangoldt function, Möbius function and Liouville function, etc. The Riesz sum of order 0 or 1 gives the well-known explicit formula for respectively the partial sum or the Riesz sum of order 1 of PNT functions. Then we may reveal the genesis of the Popov explicit formula as the integrated Davenport series with the Riesz sum of order 1 subtracted. The Fourier expansion of the Davenport series is proved to be a consequence of the functional equation, which is referred to as the Davenport expansion. By the explicit formula for the Davenport series, we also prove that the Davenport expansion for the von Mangoldt function is equivalent to the Kummer’s Fourier series up to a formula of Ramanujan and a fortiori is equivalent to the functional equation for the Riemann zeta-function.
@article{Yang2023,author={Yang, Q. and Mehta, J. and Kanemitsu, S.},title={On Popov’s Explicit Formula and the Davenport Expansion},journal={Czechoslovak Mathematical Journal},year={2023},month=mar,volume={73},pages={869-883},doi={10.21136/CMJ.2023.0322-22},note={cited By 0},url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-85152408300&doi=10.21136%2fCMJ.2023.0322-22&partnerID=40&md5=423b6d67fb760a392eeb25ab5d777b8a},publisher={Springer Science and Business Media Deutschland GmbH},issn={00114642},}
2022
Proof of the functional equation for the Riemann zeta-function
In this article, we shall prove a result which enables us to transfer from finite to infinite Euler products. As an example, we give two new proofs of the infinite product for the sine function depending on certain decompositions. We shall then prove some equivalent expressions for the functional equation, i.e. the partial fraction expansion and the integral expression involving the generating function for Bernoulli numbers. The equivalence of the infinite product for the sine functions and the partial fraction expansion for the hyperbolic cotangent function leads to a new proof of the functional equation for the Riemann zeta function.
@article{Mehta_2022,doi={10.46298/hrj.2022.7663},url={https://doi.org/10.46298%2Fhrj.2022.7663},year={2022},month=jan,publisher={Centre pour la Communication Scientifique Directe ({CCSD})},volume={44},author={Mehta, Jay and Zhu, P.-Y.},title={Proof of the functional equation for the Riemann zeta-function},journal={Hardy-Ramanujan Journal},}
Summary: In this article, we present a very short proof of infinitude of primes. This generalized proof is based on the idea of factorization of the product of primes and it unifies three proofs, the famous proof due to Euclid, a proof of Métrod, and the one due to Stieltjes.
@article{Mehta202252,author={Mehta, J.},title={A Short Generalized Proof of Infinitude of Primes},journal={College Mathematics Journal},year={2022},volume={53},number={1},pages={52-53},doi={10.1080/07468342.2022.1987114},note={cited By 0},url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-85119354300&doi=10.1080%2f07468342.2022.1987114&partnerID=40&md5=123cbe78d9696416de2a13fcd4b6e6bf},publisher={Taylor and Francis Ltd.},issn={07468342},}
2020
Some remarks on the Fourier coefficients of cusp forms
In this paper, we consider the angular changes of Fourier coefficients of half integral weight cusp forms and sign changes of q-exponents of generalized modular functions.
@article{Kumar20201935,author={Kumar, B. and Mehta, J. and Viswanadham, G.K.},title={Some remarks on the Fourier coefficients of cusp forms},journal={International Journal of Number Theory},year={2020},month=jun,volume={16},number={9},pages={1935-1943},doi={10.1142/S1793042120500992},note={cited By 0},url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-85088936227&doi=10.1142%2fS1793042120500992&partnerID=40&md5=1cf7b6f0d4c0f3cd1465bf58d3eb5b4f},publisher={World Scientific},issn={17930421},}
The boundary lerch zeta-function and short character sums À la y. Yamamoto
As has been pointed out by Chakraborty et al (Seeing the invisible: around generalized Kubert functions. Ann. Univ. Sci. Budapest. Sect. Comput. 47 (2018), 185–195), there have appeared many instances in which only the imaginary part—the odd part—of the Lerch zeta-function was considered by eliminating the real part. In this paper we shall make full use of (the boundary function aspect of) the q-expansion for the Lerch zeta-function, the boundary function being in the sense of Wintner (On Riemann’s fragment concerning elliptic modular functions. Amer. J. Math. 63 (1941), 628–634). We may thus refer to this as the ‘Fourier series–boundary q-series’, and we shall show that the decisive result of Yamamoto (Dirichlet series with periodic coefficients. Algebraic Number Theory. Japan Society for the Promotion of Science, Tokyo, 1977, pp. 275–289) on short character sums is its natural consequence. We shall also elucidate the aspect of generalized Euler constants as Laurent coefficients after a brief introduction of the discrete Fourier transform. These are rather remote consequences of the modular relation, i.e. the functional equation for the Lerch zeta-function or the polylogarithm function. That such a remote-looking subject as short character sums is, in the long run, also a consequence of the functional equation indicates the ubiquity and omnipotence of the Lerch zeta-function—and, a fortiori, the modular relation (S. Kanemitsu and H. Tsukada. Contributions to the Theory of Zeta-Functions: the Modular Relation Supremacy. World Scientific, Singapore, 2014).
@article{Wang2020313,author={Wang, X. and Mehta, J. and Kanemitsu, S.},title={The boundary lerch zeta-function and short character sums À la y. Yamamoto},journal={Kyushu Journal of Mathematics},year={2020},month=dec,volume={74},number={2},pages={313-335},doi={10.2206/kyushujm.74.313},note={cited By 3},url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-85084494022&doi=10.2206%2fkyushujm.74.313&partnerID=40&md5=f46f7661765a3f0deef855c04b494504},publisher={Kyushu University, Faculty of Science},issn={13406116},}
Boundary behavior of important functions has been an object of intensive research since the time of Riemann. Kurokawa, Kurokawa-Koyama, and Chapman studied the boundary behavior of generalized Eisenstein series which falls into this category. The underlying principle is the use of the Lipschitz summation formula. Our purpose is to show that it is a form of the functional equation for the Lipschitz-Lerch transcendent (and in the long run, it is equivalent to that for the Riemann zeta-function) and that this being indeed a boundary function of the Hurwitz-Lerch zeta-function, one can extract essential information. We also elucidate the relation between Ramanujan’s formula and automorphy of Eisenstein series.
@article{Li2020,author={Li, W. and Li, H. and Mehta, J.},title={Around the Lipschitz Summation Formula},journal={Mathematical Problems in Engineering},year={2020},month=apr,volume={2020},doi={10.1155/2020/5762823},art_number={5762823},note={cited By 2},url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-85084431836&doi=10.1155%2f2020%2f5762823&partnerID=40&md5=e4073e9aee8072a5fdda5f5ea002a43d},publisher={Hindawi Limited},issn={1024123X}}
2017
Analytic continuation of multiple Hurwitz zeta functions
We obtain the analytic continuation of multiple Hurwitz zeta functions by using a simple and elementary translation formula. We also locate the polar hyperplanes for these functions and express the residues, along these hyperplanes, as coefficients of certain infinite matrices.
@article{Mehta20171431,author={Mehta, J. and Viswanadham, G.K.},title={Analytic continuation of multiple Hurwitz zeta functions},journal={Journal of the Mathematical Society of Japan},year={2017},month=oct,volume={69},number={4},pages={1431-1442},doi={10.2969/jmsj/06941431},note={cited By 3},url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-85032792705&doi=10.2969%2fjmsj%2f06941431&partnerID=40&md5=1c860b0b160b22ba213501d389cb30bc},publisher={Mathematical Society of Japan},issn={00255645},}
2016
Analytic properties of multiple zeta functions and certain weighted variants, an elementary approach
In this article we obtain the meromorphic continuation of multiple zeta functions, together with a complete list of their poles and residues, by means of an elementary and simple translation formula for these multiple zeta functions. The use of matrices to express this translation formula leads, in particular, to a succinct description of the residues of the multiple zeta functions. We conclude our paper by introducing certain interesting weighted variants of multiple zeta functions. They are shown to behave particularly nicely with respect to product formulas and location of poles.
@article{Mehta2016487,author={Mehta, J. and Saha, B. and Viswanadham, G.K.},title={Analytic properties of multiple zeta functions and certain weighted variants, an elementary approach},journal={Journal of Number Theory},year={2016},month=nov,volume={168},pages={487-508},doi={10.1016/j.jnt.2016.04.027},note={cited By 15},url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-84975853258&doi=10.1016%2fj.jnt.2016.04.027&partnerID=40&md5=e6698d78f452e577e7f5fe2408f9ab35},publisher={Academic Press Inc.},issn={0022314X},}
In this paper, we show that any additive complex valued function over non-zero Gaussian integers which vanishes on the shifted Gaussian primes is necessarily identically zero.
@article{Mehta2015123,author={Mehta, J. and Viswanadham, G.K.},title={Set of uniqueness of shifted Gaussian primes},journal={Functiones et Approximatio, Commentarii Mathematici},year={2015},month=sep,volume={53},number={1},pages={123-133},doi={10.7169/facm/2015.53.1.7},note={cited By 0},url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-84952880100&doi=10.7169%2ffacm%2f2015.53.1.7&partnerID=40&md5=2a00bfd9ea32e736e92eb2511d2d127c},publisher={Adam Mickiewicz University Press},issn={02086573},}
2014
Quasi-uniqueness of the set of "gaussian prime plus one’s"
We recall the well-known notion of the set of uniqueness for arithmetical functions, introduced by Kátai and several other mathematicians like Indlekofer, Elliot and Hoffman, independently. We define its analogue for completely additive complex-valued functions over the set of non-zero Gaussian integers with some examples. We show that the set of «Gaussian prime plus one’s» along with finitely many Gaussian primes of norm up to some constant K is a set of uniqueness with respect to Gaussian integers. This is analogous to Kátai’s result in the case of positive integers [I. Kátai, On sets characterizing number theoretical functions, II, Acta Arith.16 (1968) 1-14].
@article{Mehta20141783,author={Mehta, J. and Viswanadham, G.K.},title={Quasi-uniqueness of the set of "gaussian prime plus one's"},journal={International Journal of Number Theory},year={2014},month=apr,volume={10},number={7},pages={1783-1790},doi={10.1142/S1793042114500559},note={cited By 1},url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-84929519532&doi=10.1142%2fS1793042114500559&partnerID=40&md5=084623f16b3449053bf219da17c7ed5d},publisher={World Scientific},issn={17930421},}
On arithmetical functions having constant values in some domain
This paper shows that any completely additive complex valued function over a principal configuration in the complex plane, having constant values in some discs, is the identically zero function. In other words, there exists no non-trivial completely additive complex valued function over a principal configuration in ℂ which assumes constant values in some domain.
@article{Mehta2014110,author={Mehta, J.},title={On arithmetical functions having constant values in some domain},journal={Acta Mathematica Hungarica},year={2014},month=aug,volume={142},number={1},pages={110-117},doi={10.1007/s10474-013-0342-8},note={cited By 0},url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-84895905737&doi=10.1007%2fs10474-013-0342-8&partnerID=40&md5=f2fbebe6021fe15560b2bd47a64a17c7},publisher={Kluwer Academic Publishers},issn={02365294},}
In this paper, we elucidate the well-known Wilton’s formula for the product of two Riemann zeta functions. A proof of Wilton’s expression for product of two zeta functions was given by M. Nakajima in [5] using the Atkinson dissection. On the similar line we derive Wilton’s formula using the Riesz sum of the order κ = 1 \(k=1\).
@article{Banerjee2014123,author={Banerjee, D. and Mehta, J.},title={Linearized product of two Riemann zeta functions},journal={Proceedings of the Japan Academy Series A: Mathematical Sciences},year={2014},month=oct,volume={90},number={8},pages={123-126},doi={10.3792/pjaa.90.123},note={cited By 2},url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-84908243164&doi=10.3792%2fpjaa.90.123&partnerID=40&md5=bf7d1ad664534b7956fe8717a22c0faa},publisher={Japan Academy},issn={03862194},}
Preventing Unknown Key-Share Attack using Cryptographic Bilinear Maps
Here we add a third pass to the two pass AK protocols, MTI/AO protocol and a two-pass protocol proposed by L. Law et al. using cryptographic bilinear maps. The added third pass provides additional key confirmation and also prevents the unknown key-share attack which could be successfully launched on the above mentioned two-pass protocols.
@article{Chakraborty2014135,author={Chakraborty, K. and Mehta, J.},title={Preventing Unknown Key-Share Attack using Cryptographic Bilinear Maps},journal={Journal of Discrete Mathematical Sciences and Cryptography},year={2014},month=aug,volume={17},number={2},pages={135-147},doi={10.1080/09720529.2013.842035},note={cited By 2},url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-84905441696&doi=10.1080%2f09720529.2013.842035&partnerID=40&md5=f3c70776733b9356a9b88e61abe761a9},publisher={Taru Publications},issn={09720529},}
2012
A stamped blind signature scheme based on elliptic curve discrete logarithm problem
Here we present a stamped blind digital signature scheme which is based on elliptic curve discrete logarithm prob-lem and collision-resistant cryptographic hash functions.
@article{Chakraborty2012316,author={Chakraborty, K. and Mehta, J.},title={A stamped blind signature scheme based on elliptic curve discrete logarithm problem},journal={International Journal of Network Security},year={2012},month=nov,volume={14},number={6},pages={316-319},note={cited By 8},url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-84872790760&partnerID=40&md5=7a36c47407f15b53c67273cec62bad17},issn={1816353X},}
Preprints
Book Chapters
2023
Chapter 18: On Periodic Dirichlet Series and Special Functions, Advanced Mathematical Analysis and its Applications (1st ed.).
We present an example of algebraic elucidation of analytic expressions by the finite form of the Dirichlet class number formula. It was shown by [Hashimoto et al. (2008)] that the finite form for the Dirichlet \(L\)-function value \(L(1,χ)\), \(χ≠\chi_0\) is equivalent to Gauss’ first formula for the Euler digamma function. In this paper, we shall generalize this to the case of Discrete Fourier Transform and reveal the underlying algebraic structure.
@book{mehtakataikanemitsu,title={Chapter 18: On Periodic Dirichlet Series and Special Functions, Advanced Mathematical Analysis and its Applications (1st ed.).},author={Mehta, Jay and Kátai, Imre and Kanemitsu, S.},year={2023},publisher={Chapman and Hall/CRC},doi={https://doi.org/10.1201/9781003388678},}
