\end{cases}\displaystyle \end{aligned}$$, $$ \sum_{-n/2< r \leq n/2} \Biggl(\sum _{ \substack{k=-\infty \\ k \equiv r (\operatorname{mod} n)}}^{\infty } a^{{k(k+1)}/{(2n)}}b^{{k(k-1)}/{(2n)}} \Biggr)^{n}=f(a,b)F_{n}(ab), $$, $$ F_{n}(q):=1+2nq^{(n-1)/2}+\cdots , \quad n \geq 3. [24][25][26] In the primary literature, the series 1 + 2 + 3 + 4 + is mentioned in Euler's 1760 publication De seriebus divergentibus alongside the divergent geometric series 1 + 2 + 4 + 8 + . continues to hold when both functions are extended by analytic continuation to include values of s for which the above series diverge. 1, pp. and elementary trigonometric identities. Forum Math. [1] In addition to the expansions discussed in this article, Ramanujan's sums are used in the proof of Vinogradov's theorem that every sufficiently large odd number is the sum of three primes. n Math. Lithuanian Math J 62(2):274284, Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok, 10900, Thailand, Office of the President, King Mongkuts University of Technology Thonburi, Ratchaburi Learning Park, Ratchaburi, 70150, Thailand, You can also search for this author in {\displaystyle a\mid b} $$, $$ \sum_{k=0}^{n-1}\prod _{j=1}^{n}\vartheta _{1} \biggl(z+y_{j}+ \frac{k \pi }{n} \Big|\tau \biggr)=R_{1,1}^{(1)} (y_{1}, y_{2}, \ldots , y_{n};n,p;\tau ) \vartheta _{3} (nz|n \tau ), $$, $$ R_{1,1}^{(1)} (y_{1}, y_{2}, \ldots , y_{n};n,p;\tau )=(-1)^{p}nq^{- \frac{n}{4}} \sum_{ \substack{r_{1}, \ldots , r_{n}=-\infty \\ r_{1}+\cdots +r_{n}=\frac{n}{2}}}^{\infty }q^{r^{2}_{1}+\cdots +r^{2}_{n}}e^{-2i (r_{1}y_{1}+\cdots +r_{n}y_{n} )}. Integral Transforms Spec. : Ramanujans Notebooks Part III. \end{aligned}$$, \(R_{1,1}^{(1)} (m,n;\tau )=R_{1,1}^{(2)} (m,n;\tau )=2m \vartheta _{2} (0| 2 \tau )\), $$\begin{aligned} & \sum_{k=0}^{4m-1} \vartheta _{1}^{4} \biggl(z+\frac{k \pi }{4m} \Big| \tau \biggr)=R(m;\tau )\vartheta _{3} \bigl(4mz|4m^{2} \tau \bigr), \end{aligned}$$, $$\begin{aligned} & \sum_{k=0}^{4m-1} \vartheta _{1}^{4} \biggl(z+\frac{\pi }{8m}+ \frac{k \pi }{4m} \Big|\tau \biggr)=R(m;\tau )\vartheta _{4} \bigl(4mz|4m^{2} \tau \bigr), \end{aligned}$$, $$ R(m;\tau )=4mq^{-1} \sum _{ \substack{r_{1}, \ldots , r_{4}=-\infty \\ r_{1}+\cdots +r_{4}=2}}^{\infty }q^{r^{2}_{1}+\cdots +r^{2}_{4}}. a The work of D. Lim was partially supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean government (MSIT) NRF-2021R1C1C1010902. 20, no. Dongkyu Lim. your institution. MATH Google Scholar, Andrews, G.E., Berndt, B.C. Language links are at the top of the page across from the title. Springer Nature or its licensor (e.g. This process involves a bit of mathematical wizardry, but it is a powerful tool in a variety of mathematical fields. When \(ax+by=\frac{(2p+1)\pi }{2m}\), we have, Setting \(y_{1}=y_{2}=\cdots =y_{a}=x\) and \(y_{a+1}=y_{a+2}=\cdots =y_{n}=y\) with \(a+b=n\) in Theorem2.1, we obtain Corollary4.9, Taking \(p=0\), \(a=b=1\) in Corollary4.9, we have, and noting that \(x+y=0\) and \(x+y=\frac{\pi }{2m}\), we obtain Corollary4.10. in equation (2.1) of Theorem2.1 and applying properties (1.9) and (1.19), we arrive at formula (2.16) of Theorem2.2. in equation (2.3) of Theorem2.1 and applying properties (1.9) and (1.20), we arrive at formula (2.18) of Theorem2.2. WebEven though Ramanujan did find the "sum" -1/12 for this series, he did so in a way that is valid and useful in specific contexts and which produces the correct sums for convergent series, indicating that his method is well-defined. All authors read and approved the final manuscript. Provided by the Springer Nature SharedIt content-sharing initiative, https://doi.org/10.1007/s11139-023-00733-1, access via The present chapter discusses Ramanujan Sums and its various signal processing applications. 2, pp. MathSciNet The cutoff function should have enough bounded derivatives to smooth out the wrinkles in the series, and it should decay to 0 faster than the series grows. By accepting all cookies, you agree to our use of cookies to deliver and maintain our services and site, improve the quality of Reddit, personalize Reddit content and advertising, and measure the effectiveness of advertising. The divergence is a simple consequence of the form of the series: the terms do not approach zero, so the series diverges by the term test. As a function of z, show that fis holomorphic in the Equating the constants of both sides of (2.13) and noting the condition \(y_{1}+y_{2}+\cdots +y_{n}=\frac{p \pi }{m}\), we get (2.2). s ": (a;q),~x"_= aOt(. where = 0.5772 is the EulerMascheroni constant. Claims of John McAfee faked his own death stirs Internet. q As Ruth launches into a derivation of the functional equation of the zeta function, another actor addresses the audience, admitting that they are actors: "But the mathematics is real. A note on certain summations due to Ramanujan with application and generalization. MATH \end{aligned}$$, $$\begin{aligned} & \sum_{k=0}^{2m-1} \vartheta _{1} \biggl(z+x+\frac{k \pi }{2m}\Big| \tau \biggr) \vartheta _{1} \biggl(z-x+\frac{k \pi }{2m} \Big|\tau \biggr)=2m \vartheta _{2} (2x|2 \tau )\vartheta _{3} \bigl(2mz|2m^{2} \tau \bigr), \end{aligned}$$, $$\begin{aligned} & \sum_{k=0}^{2m-1} \vartheta _{1} \biggl(z+x+\frac{k \pi }{2m} \Big| \tau \biggr) \vartheta _{1} \biggl(z-x+\frac{(k+1) \pi }{2m} \Big|\tau \biggr) \\ &\quad =2m \vartheta _{2} \biggl(2x- \frac{\pi }{2m}\Big|2 \tau \biggr) \vartheta _{4} \bigl(2mz|2m^{2} \tau \bigr). Provided by the Springer Nature SharedIt content-sharing initiative, Mathematical and Computational Intelligence to Socio-scientific Analytics and Applications, https://doi.org/10.1007/978-981-19-5181-7_6. ( 7, pp. a For n is even, hence \(\frac{mn}{2}\) is a positive integer. and our 79, 183187 (1997), Article q Haukkanen P., Discrete Ramanujan Fourier transform of even functions (mod r), Indian J. 126. ( On page 54 in Ramanujans lost notebook (see [21, p.54, Entry 9.1.1], [2, p.337]), Ramanujan recorded the following claim (without proof), which is now well known as Ramanujans circular summation. \end{aligned}$$, $$\begin{aligned}& e^{(2p+1)iz} \vartheta _{1}^{2} \biggl(z+p \pi \tau + \frac{\pi \tau }{2}\Big|2\tau \biggr)+q^{2p+2}e^{(2p+3)iz} \vartheta _{1}^{2} \biggl(z+p \pi \tau + \frac{3\pi \tau }{2}\Big|2\tau \biggr) \\& \quad =(-1)^{p}q^{-\frac{1}{4}} \bigl[q^{-p}\vartheta _{2}(2p \pi \tau |4\tau )-q^{p+1} \vartheta _{2} \bigl((2p+1) \pi \tau |4\tau \bigr)\vartheta _{2}(z|\tau ) \bigr]. Google Scholar, Berndt, B.C. The generating functions of the Ramanujan sums are Dirichlet series: is a generating function for the sequence cq(1), cq(2), where q is kept constant, and. 20, pp. n The values of the Riemann zeta function at the positive even integers are related to the Bernoulli numbers, and Ramanujan summation is used to evaluate these values. Web1 The short of it is that Ramanujan's summation involves a certain manipulation that isn't quite immediate. ( Suppose that n is even; \(y_{1}, y_{2}, \ldots , y_{n}\) are any complex numbers. Ramanujan found expansions of some of the well-known functions of number theory. ( In: Proceedings of IEMCI-2019, Jabalpur, McCarthy PJ (1986) Introduction to arithmetical functions. 7580, 2007. \end{aligned}$$, \(\tau \longmapsto -\frac{1}{m^{2}n \tau }\), \(y_{j} \longmapsto \frac{y_{j} \pi }{m^{2}n}\), $$\begin{aligned}& \sum_{k=0}^{mn-1}\prod _{j=1}^{n}\vartheta _{1} \biggl( \frac{mz+y_{j} \pi \tau +mk \pi \tau }{m^{2}n \tau } \Big|{-}\frac{1}{m^{2}n \tau } \biggr) \\& \quad =R_{1,1}^{(1)} \biggl(\frac{y_{1} \pi }{m^{2}n}, \ldots , \frac{y_{n} \pi }{m^{2}n};m,n,p;- \frac{1}{m^{2}n \tau } \biggr)\vartheta _{3} \biggl( \frac{z}{\tau } \Big|{-}\frac{1}{\tau } \biggr). n In the same publication, Euler writes that the sum of 1 + 1 + 1 + 1 + is infinite. Soc., vol. Setting \(p=0\) in Theorem2.1, we have the following. J. These keywords were added by machine and not by the authors. N Abel summation is a more powerful method that not only sums Grandi's series to 1/2, but also sums the trickier series 1 2 + 3 4 + to 1/4. : On series for \(1/\pi \) and \(1/\pi ^2\). \end{aligned}$$, $$\begin{aligned} & \sum_{k=0}^{mn-1}q^{k^{2}}e^{2kiz} \vartheta _{1}^{n} \bigl(mz+mk \pi \tau |m^{2}n \tau \bigr)=F_{1,1}^{(1)} (m,n; \tau )\vartheta _{3} (z|\tau ), \end{aligned}$$, $$\begin{aligned} & \sum_{k=0}^{mn-1}q^{k^{2}+k}e^{(2k+1)iz} \prod_{j=1}^{n}\vartheta _{1} \biggl(mz+\frac{m \pi \tau }{2}+mk \pi \tau |m^{2}n \tau \biggr)=F_{1,1}^{(2)} (m,n;\tau )\vartheta _{2} (z| \tau ), \end{aligned}$$, $$\begin{aligned} & F_{1,1}^{(1)} (m,n;\tau )= \frac{(\sqrt{-i})^{1-3n}\sqrt{\tau ^{1-n}}}{(m \sqrt{n})^{n}} R_{1,1}^{(1)} \biggl(m,n;- \frac{1}{m^{2}n\tau } \biggr), \end{aligned}$$, $$\begin{aligned} & F_{1,1}^{(1)} (m,n;\tau )= i^{n} \sum_{k=0}^{mn-1}q^{- (k+\frac{mn}{2} )^{2}} \sum_{ \substack{r_{1}, \ldots , r_{n}=-\infty \\ 2m(r_{1}+\cdots +r_{n})=mn+2k}}^{\infty }(-1)^{r_{1}+\cdots +r_{n}}q^{m^{2}n(r_{1}^{2}+ \cdots +r_{n}^{2 })}, \end{aligned}$$, $$\begin{aligned} & F_{1,1}^{(2)} (m,n;\tau )= \frac{(\sqrt{-i})^{1-3n}\sqrt{\tau ^{1-n}}}{(m \sqrt{n})^{n}} q^{- \frac{1}{4}} R_{1,1}^{(2)} \biggl(m,n;-\frac{1}{m^{2}n\tau } \biggr), \end{aligned}$$, $$\begin{aligned} & F_{1,1}^{(2)} (m,n;\tau )=i^{n} \sum_{k=0}^{mn-1}q^{- (k+\frac{mn}{2}+\frac{1}{2} )^{2}} \sum_{ \substack{r_{1}, \ldots , r_{n}=-\infty \\ 2m(r_{1}+\cdots +r_{n})=mn+2k}}^{\infty }(-1)^{r_{1}+\cdots +r_{n}}q^{m^{2}n(r_{1}^{2}+ \cdots +r_{n}^{2 })}. } WebRamanujan'ssum isa usefulextension ofJacobi's tripleproduct formula, andhas recentlybecomeimportant inthetreatmentofcertain orthogonal p lynomials definedbybasic hypergeometricseries. In the present section, we give some special cases of Theorem2.1 and derive some theta function identities. s It is a fact[3] that the powers of q are precisely the primitive roots for all the divisors of q. is the sum of the n-th powers of all the roots, primitive and imprimitive, It follows from the identity xq 1 = (x 1)(xq1 + xq2 + + x + 1) that, This shows that cq(n) is always an integer. Number Theory 8, 19772002 (2012), Luo, Q.-M.: A note for Ramanujans circular summation formula. \end{aligned}$$, $$\begin{aligned} & \vartheta _{1} (z+n \pi |\tau )=(-1)^{n} \vartheta _{1} (z|\tau ),\qquad \vartheta _{1} (z+n \pi \tau | \tau )=(-1)^{n}q^{-n^{2}}e^{-2niz} \vartheta _{1} (z|\tau ), \end{aligned}$$, $$\begin{aligned} & \vartheta _{2} (z+n \pi |\tau )=(-1)^{n} \vartheta _{2} (z|\tau ),\qquad \vartheta _{2} (z+n\pi \tau |\tau )=q^{-n^{2}}e^{-2niz} \vartheta _{2} (z|\tau ), \end{aligned}$$, $$\begin{aligned} & \vartheta _{3} (z+n \pi |\tau )=\vartheta _{3} (z| \tau ),\qquad \vartheta _{3} (z+n\pi \tau | \tau )=q^{-n^{2}}e^{-2niz}\vartheta _{3} (z|\tau ), \end{aligned}$$, $$\begin{aligned} & \vartheta _{4} (z+n \pi |\tau )=\vartheta _{4} (z| \tau ),\qquad \vartheta _{4} (z+n\pi \tau | \tau )=(-1)^{n}q^{-n^{2}}e^{-2niz}\vartheta _{4} (z|\tau ). Setting \(p=0\) in Corollary4.7, we have the following. Passionate Author. Whatever the "sum" of the series might be, call it c = 1 + 2 + 3 + 4 + . 269, pp. Here are some of the notable applications: In quantum field theory, Ramanujan summation is used to regularize the divergent integrals that arise in the perturbative expansion of quantum field theories. For an extreme example, appending a single zero to the front of the series can lead to a different result.[1]. Number Theory 48, 364372 (1994), Shen, L.C. It was brought XXII, no. \end{aligned}$$, https://doi.org/10.1186/s13662-020-03115-9, Topics in Special Functions and q-Special Functions: Theory, Methods, and Applications, http://creativecommons.org/licenses/by/4.0/. For \(q=e^{\pi {i}\tau }\), \(\operatorname{Im} (\tau )>0\), \(z \in \mathbb{C}\). : Ramanujans Notebooks Part V. Springer, New York (1998), Boon, M., Glasser, M.L., Zak, J., Zucker, I.J. Mobius function (n) is a number-theoretic function and is defined as, Here, p>2 is a prime and \(p|n\) implies p divides n. The value of twin prime constant, \(C_{2}\) is 0.660. Do give a try reading my book-10 Days: Game Over. $$, $$\begin{aligned} & \sum_{k=0}^{2m-1} \vartheta _{1}^{2} \biggl(z+\frac{k \pi }{2m} \Big|\tau \biggr)=2m \vartheta _{2} (0|2 \tau ) \vartheta _{3} \bigl(2mz|2m^{2} \tau \bigr), \end{aligned}$$, $$\begin{aligned} & \sum_{k=0}^{2m-1} \vartheta _{1}^{2} \biggl(z+\frac{\pi }{4m}+ \frac{k \pi }{2m} \Big|\tau \biggr)=2m \vartheta _{2} (0|2 \tau ) \vartheta _{4} \bigl(2mz|2m^{2} \tau \bigr). Privacy Policy. Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. 3, no. Therefore, since there are q of them, they are all of the roots. [2], For integers a and b, where (a, q) = 1. x Planat M., Ramanujan sums for signal processing of low frequency noise, in: Frequency Control Symposium and PDA Exhibition, 2002. is read "a does not divide b". only using formal manipulations of series and the simplest results about convergence). From the definition (or Kluyver's formula) it is straightforward to prove that, if p is a prime number, This result and the multiplicative property can be used to prove, This is called von Sterneck's arithmetic function. : Circular summation of the 13th powers of Ramanujans theta function. ( By (1.8), (2.7), and (2.14), we find that the function \(\frac{f(z)}{\vartheta _{4}(z|\tau )}\) is an elliptic function with double periods and , and has only a simple pole at \(z=\frac{\pi \tau }{2}\) in the period parallelogram. . DeceasedRichard B. Paris, Passed away on July 8, 2022. 2 {\displaystyle \zeta (s)} According to Morris Kline, Euler's early work on divergent series relied on function expansions, from which he concluded 1 + 2 + 3 + 4 + = . J. 211216. The divergent integrals are first expressed as infinite sums, and then Ramanujan summation is used to assign a finite value to these sums. [1, 2, 411, 13, 14, 16, 1820, 2224, 26, 27, 29, 30]). Springer, New York (2012), Book Abstract. volume2020, Articlenumber:690 (2020) AI content generation: free ai content generation tools creator needs, Are we still Running with the devil? n We have, Comparing (2.5) and (2.6), when n is only even, we have, By (1.13) and noting that n is even, we obtain, When \(y_{1}+y_{2}+\cdots +y_{n}=\frac{p \pi }{m}\) in (2.8), we have, We construct the function \(\frac{f(z)}{\vartheta _{3}(z|\tau )}\). Keywords:divergent series; summation methods; EulerMaclaurin summation formula; Ramanujan summation; fractional nite sum 1. , Putting \(n=2\) in Theorem4.2 and noting that \(R_{1,1}^{(1)} (m,n;\tau )=R_{1,1}^{(2)} (m,n;\tau )=2m \vartheta _{2} (0| 2 \tau )\), we get Corollary4.3. 29, no. Phys. From Theorem3.1 and Theorem3.2 we may obtain more theta function identities. MathSciNet {\displaystyle \phi (n)} Note that the constant is the inverse[18] of the one in the formula for (n). 2014, 243 (2014), Zhu, J.-M.: An alternate circular summation formula of theta functions and its applications. Jpn. [31] The 8-minute video is narrated by Tony Padilla, a physicist at the University of Nottingham. i {\displaystyle r'_{2s}(n)} 126129. The sum (1) where runs through the residues relatively prime to , which is important in the representation of numbers by the sums of squares. [27], David Leavitt's 2007 novel The Indian Clerk includes a scene where Hardy and Littlewood discuss the meaning of this series. Value Probl. Accordingly, Ramanujan writes. Ramanujans circular summation is an interesting subject in his notebook. The first author was partially supported by National Security Agency grant MD A904-99- 1-0003 . Ramanujan recorded his now famous 1 1 summation as item 17 of Chapter 16 in the second of his three notebooks [13, p. 32], [46]. Number Theory 130, 11901196 (2010), Chandrasekharan, K.: Elliptic Functions. [19][20], r2s(n) is the number of way of representing n as the sum of 2s squares, counting different orders and signs as different (e.g., r2(13) = 8, as 13 = (2)2 + (3)2 = (3)2 + (2)2.). 1, pp. Math. Planat M., Minarovjech M.and Saniga M., Ramanujan sums analysis of long-period sequences and 1/f noise, EPL J.; https://doi.org/10.1209/0295-5075/85/400052008. Bull,, vol. \\ & \psi_{2} \left( n \right) = n^{2} \mathop \prod \limits_{i} \left( {1 - \frac{1}{{n_{i}^{2} }}} \right) \\ \end{aligned}$$, $$\begin{array}{*{20}l} {\mu \left( n \right)} \hfill & { = 1} \hfill & {i{\text{f}}\;n = 1} \hfill \\ {} \hfill & { = \left( { - 1} \right)^{k} } \hfill & {{\text{if}}\;n = p_{1} p_{2} \ldots .p_{k} } \hfill \\ {} \hfill & { = 0} \hfill & {\text{otherwise}} \hfill \\ \end{array}$$, $$\begin{array}{*{20}c} { \wedge \left( n \right)} & = & {\{ \ln p\quad {\text{if}}\;n = p^{\beta } ,\;p\;\;{\text{is prime}}} \\ {} & = & {\text{Otherwise}} \\ \end{array}$$, $$\begin{array}{*{20}l} {C\left( n \right)} \hfill & { = 2C_{2} \mathop \prod \limits_{p|n} \frac{p - 1}{p - 2},} \hfill & {{\text{if}}\;n\,{\text{is odd}}} \hfill \\ {} \hfill & { = 0,} \hfill & {{\text{if}}\;n\,{\text{is even}}} \hfill \\ \end{array}$$, https://doi.org/10.1007/978-981-10-8234-4_34, https://doi.org/10.1209/0295-5075/85/400052008. Sci. Apostol T.M., Arithmetical properties of generalized Ramanujan Sums, Pacific J. of Mathematics, vol 41, No. Soc., 1932, pp. The infinite series whose terms are the natural numbers 1 + 2 + 3 + 4 + is a divergent series. Am. the numbers 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, 15, ; the n-th triangular number is given by the formula n(n + 1)/2. Instead, such a series must be interpreted by zeta function regularization. Tax calculation will be finalised at checkout. They also proved that Theorem1.9 is equivalent to Theorem1.7. Cohen E., An extension of Ramanujans sum. Luo [17] further generalized the results of Chan and Liu on Ramanujans circular summation formula for theta functions \(\vartheta _{3} (z|\tau )\) and deduced some alternating summation formulas of theta functions \(\vartheta _{1} (z|\tau )\) and \(\vartheta _{2} (z|\tau )\). This can be seen as follows. J. k Scan this QR code to download the app now. Recently, Liu and Luo [15] obtained the alternating circular summation formulas of theta function \(\vartheta _{3} (z|\tau )\). ( ) s J. Comput. 7, discusses Ramanujan expansions as a type of Fourier expansion in an inner product space which has the, Cf. Different methods may yield different values for the same series, and in some cases, the assigned value may not have a clear physical or mathematical interpretation. Gen. 15, 34393440 (1982), Borwein, J.M., Borwein, P.B. Proc. If f(n) is an arithmetic function (i.e. , Taking \(m=1\) in Corollary4.10, we have, Taking \(m=2\) in Corollary4.10, we have. https://doi.org/10.1186/s13662-020-03115-9, DOI: https://doi.org/10.1186/s13662-020-03115-9. In the same manner, using the imaginary transformations formulas, to (2.3) and noting that \(y_{1}+y_{2}+\cdots +y_{n}=\frac{(2p+1)mn}{2}\), we can prove formulas (3.4), (3.5), and (3.6), respectively. Also let \(y_{1}, y_{2}, \ldots , y_{n}\) be any complex numbers. : Circular summation of theta functions in Ramanujans Lost Notebook. ) This proof is complete. Pei S.-C. and Chang K.-W., Odd Ramanujan sums of complex roots of unity, IEEE Signal Process. {\displaystyle \{c_{q}(1),c_{q}(2),\ldots \}} J. Integral Transforms Spec. b In the following formulas the signs repeat with a period of 4. r e Learn more about Institutional subscriptions, Andrews, G.E., Askey, R., Roy, R.: Special Functions. Soc. Dividing both sides by 3, one gets c=+1/12. WebSOME APPLICATIONS OF RAMANUJAN'S TRIGONOMETRICAL SUM Cm (n) BY K. G. RAMANATHAN* Unirersity of Madras Received May 20, 1944 (Communicated by Dr. R. Vaidyanathaswamy, F.A.Sc.) London Math. In the third section we derive the corresponding imaginary transformation formulas of circular summation formulas by using the imaginary transformation formulas of \(\vartheta _{1}(z|\tau )\) and \(\vartheta _{2}(z|\tau )\). Srinivasa Ramanujan presented two derivations of "1 + 2 + 3 + 4 + = +1/12" in chapter 8 of his first notebook. WebThe main contribution of this paper is to propose a closed expression for the Ramanujan constant of alternating series, based on the EulerBoole summation formula. ( 2013, 59 (2013), Chan, H.H., Liu, Z.-G., Ng, S.T. J. See Ya! n Cite this article. ) [7][8][9] The simpler, less rigorous derivation proceeds in two steps, as follows. Ramanujan refers to the same paper as he did for the squares, where he showed that there is a function x $$, $$ f(a,b)=\sum_{n=-\infty }^{\infty }a^{{n(n+1)}/{2}}b^{{n(n-1)}/{2}},\quad \vert ab \vert < 1. For example, you can use Ramanujan summation to assign a finite value to the infinite series [Math Processing Error], which we know diverges. $$, \(y_{1}+y_{2}+\cdots +y_{n}=\frac{p \pi }{m}\), $$\begin{aligned} \sum_{k=0}^{mn-1}\prod _{j=1}^{n}\vartheta _{1} \biggl(z+y_{j}+ \frac{k \pi }{mn}\Big|\tau \biggr)=R_{1,1}^{(1)} (y_{1}, y_{2}, \ldots , y_{n};m,n,p;\tau ) \vartheta _{3} \bigl(mnz|m^{2}n \tau \bigr), \end{aligned}$$, $$ R_{1,1}^{(1)} (y_{1}, y_{2}, \ldots , y_{n};m,n,p;\tau )=mnq^{- \frac{n}{4}} e^{\frac{p \pi {i}}{m}}\sum_{ \substack{r_{1}, \ldots , r_{n}=-\infty \\ r_{1}+\cdots +r_{n}=\frac{n}{2}}}^{\infty }q^{r^{2}_{1}+\cdots +r^{2}_{n}}e^{-2i (r_{1}y_{1}+\cdots +r_{n}y_{n} )}. Unknown Facts About The Math Wizard Srinivasa Ramanujan. Math. The results so obtained complement our earlier work in the Dirichlet convolution setting. 2 ( c ) Gordon and Breach Science Publishers, Amsterdam (1990), Rathie, A.K., Rakha, M.A. The nth partial sum of the series is the triangular number. \end{aligned}$$, $$\begin{aligned} & \vartheta _{1} \biggl(z+{\frac{\pi }{2}}\Big| \tau \biggr)=\vartheta _{2} (z|\tau ),\qquad \vartheta _{1} \biggl(z+\frac{\pi \tau }{2}\Big|\tau \biggr)=iq^{-\frac{1}{4}}e^{-iz} \vartheta _{4} (z| \tau ), \end{aligned}$$, $$\begin{aligned} & \vartheta _{2} \biggl(z+\frac{\pi }{2}\Big| \tau \biggr)=-\vartheta _{1}(z| \tau ),\qquad \vartheta _{2} \biggl(z+\frac{\pi \tau }{2}\Big|\tau \biggr)=q^{-\frac{1 }{4}}e^{-iz} \vartheta _{3} (z|\tau ), \end{aligned}$$, $$\begin{aligned} & \vartheta _{3} \biggl(z+\frac{\pi }{2}\Big| \tau \biggr)=\vartheta _{4}(z| \tau ),\qquad \vartheta _{3} \biggl(z+\frac{\pi \tau }{2}\Big|\tau \biggr)=q^{-\frac{1 }{4}}e^{-iz} \vartheta _{2} (z|\tau ), \end{aligned}$$, $$\begin{aligned} & \vartheta _{4} \biggl(z+\frac{\pi }{2}\Big| \tau \biggr)=\vartheta _{3}(z| \tau ),\qquad \vartheta _{4} \biggl(z+\frac{\pi \tau }{2}\Big|\tau \biggr)=iq^{-\frac{1 }{4}}e^{-iz} \vartheta _{1} (z|\tau ). This chapter clearly articulates the usage and importance of Ramanujan Sums in a number of signal processing aspects. Math. Google Scholar, Department of Mathematics, Vedant College of Engineering & Technology (Rajasthan Technical University), Bundi, Rajasthan, 323021, India, Department of Mathematics Education, Andong National University, Andong, 36729, Republic of Korea, Division of Computing and Mathematics, University of Abertay, Dundee, UK, You can also search for this author in (2019). When \(y_{1}+y_{2}+\cdots +y_{n}=\frac{\pi }{2m}\), we have, Suppose that n is even, m is any positive integer, then, Taking \(y_{1}=y_{2}=\cdots =y_{n}\) in Corollary4.1, we get Corollary4.2. Number Theory 132, 11641169 (2012). It is easily shown from the definition that cq(n) is multiplicative when considered as a function of q for a fixed value of n:[5] i.e. Hope you have enjoyed the read. Connections with totient functions, Duke Math. 715720. VLSI architectures to calculate Ramanujan Sums and DFT These relationships can be expressed using algebra. AMS Chelsea Publishing, Cambridge University Press, Cambridge (1940), Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I. [6] The equivalence of it and Ramanujan's sum is due to Hlder.[7][8]. The present investigation was supported by the Natural Science Foundation General Project of Chongqing, China under grant cstc2019jcyj-msxmX0143 and the Science and Technology Research Program of Chongqing Municipal Education Commission, China under grant No. on Signal Proc., vol. , Similarly, by applying the imaginary transformations formulas. Correspondence to 2 41584172, Aug., 2014. Laohakosol V., Ruengsinsub P. and Pabhapote N., Ramanujan Sums for signal processing of low frequency noise, Phys. (This can be seen by equating 1/1 + x to the alternating sum of the nonnegative powers of x, and then differentiating and negating both sides of the equation.) Amer Math Monthly 78(5):545546, CrossRef (eds) Proceeding of the Second International Conference on Microelectronics, Computing & Communication Systems (MCCS 2017). Ramanujan summation involves taking an infinite series and assigning it a value based on an analytical continuation of a related function. One can then prove that this smoothed sum is asymptotic to +1/12 + CN2, where C is a constant that depends on f. The constant term of the asymptotic expansion does not depend on f: it is necessarily the same value given by analytic continuation,+1/12.[1]. Int. In: Nath, V., Mandal, J. PubMedGoogle Scholar. = Cohen L., Time-frequency distributionsA review, Proc. The discussion might be a bit above your head but I had a similar question last month and looking at it might be fruitful to you, especially regarding the discussion of how he derives it: math.stackexchange.com/questions/3044810/ , is also a root. < The first key insight is that the series of positive numbers 1 + 2 + 3 + 4 + closely resembles the alternating series 1 2 + 3 4 + . Provided by the Springer Nature SharedIt content-sharing initiative, Some Ramanujan-type circular summation formulas, $$\begin{aligned}& \vartheta _{1} (z|\tau ) =-iq^{\frac{1}{4}}\sum _{n=-\infty }^{\infty } (-1 )^{n}q^{n ( n+1 )}e^{ (2n+1 )iz}, \end{aligned}$$, $$\begin{aligned}& \vartheta _{2} (z|\tau ) =q^{\frac{1}{4}}\sum _{n=-\infty }^{ \infty } q^{n (n+1 )}e^{ (2n+1 )iz}, \end{aligned}$$, $$\begin{aligned}& \vartheta _{3} (z|\tau ) =\sum _{n=-\infty }^{\infty }q^{n^{2}}e^{2niz}, \end{aligned}$$, $$\begin{aligned}& \vartheta _{4} (z|\tau ) =\sum _{n=-\infty }^{\infty } (-1 )^{n}q^{n^{2}}e^{2niz}. \end{aligned}$$, \(C_{1,1}^{(1)}(y_{1}, y_{2}, \ldots , y_{n};\tau )\), $$ f(z)=C_{1,1}^{(1)}(y_{1}, y_{2}, \ldots , y_{n};\tau )\vartheta _{3}(z| \tau ), $$, $$ \sum_{k=0}^{mn-1}\prod _{j=1}^{n}\vartheta _{1} \biggl(\frac{z}{mn}+y_{j}+ \frac{k\pi }{mn} \Big| \frac{\tau }{m^{2}n} \biggr)=C_{1,1}^{(1)} (y_{1}, y_{2}, \ldots , y_{n};\tau ) \vartheta _{3} (z| \tau ). 0(n), the number of divisors of n, is usually written d(n) and 1(n), the sum of the divisors of n, is usually written (n). Monatsh Math 70:149154, Schramm W (2008) The Fourier transform of functions of the greatest common divisor. Number Theory 76, 6265 (1999), Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. The eta function is defined by an alternating Dirichlet series, so this method parallels the earlier heuristics. q Am. Cambridge University Press, Cambridge (2000), MATH n https://doi.org/10.1007/978-981-19-5181-7_6, DOI: https://doi.org/10.1007/978-981-19-5181-7_6, eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0). 843851. The latter series is an example of a Dirichlet series. And Breach Science Publishers, Amsterdam ( 1990 ), Prudnikov, A.P., Brychkov, Yu.A., Marichev O.I... Book-10 Days: Game Over inthetreatmentofcertain orthogonal p lynomials definedbybasic hypergeometricseries ( 2010 ), Shen L.C. 2008 ) the Fourier transform of functions of number Theory 8, 2022 may obtain more function... 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