Professor (Mathematics)

Ph. D. in Mathematics from the Institute of Mathematical Sciences, Madras (Degree awarded by the University of Madras).
Thesis supervisor : Prof. R. Balasubramanian.

Mathematical Interests : Number Theory and Combinatorics.

#### Research Papers:

SL NO

Title Authorship Publication
1 On an error term related to the greatest divisor of $n$, which is prime to $k$. (Written jointly with R. Balasubramanian and A. Sankaranarayanan) Indian J. Pure Appl. Math. (9) 19, 830-841 (1988).
2
An $\Omega$- result related to $r_4(n)$.
(with R. Balasubramanian and A. Sankaranarayanan) Hardy - Ramanujan Journal 12, 20-30 (1989).
3
On an error term related to the Jordan totient function $J_k(n)$.
(Written jointly with A. Sankaranarayanan) J. Number Theory 34, no 2., 178-188 (1990).
4
A note on a certain error term.
(Written jointly with R. Balasubramanian) Arch. Math. 56, 37-40 (1991).
5
$\Omega$- results for sums of Fourier coefficients of cusp forms.
- Acta Arithmetica 57, no. 2, 83-92 (1991)
6 Lattice points in ellipsoids. (Written jointly with Y. -F. S. P\'etermann) Acta Arithmetica 59, no. 4, 329-338 (1991)
7
Towards the exact nature of a certain error term.
- Arch. Math. 58, 257-264 (1992).
8
Towards the exact nature of a certain error term -II.
(Written jointly with K. Soundararajan) Arch. Math. 59, 442-449 (1992).
9
On the average behaviour of an arithmetical function.
(Written jointly with K. Chakraborty) Arch. Math. 62, 411-417 (1994.
10
A note on the distribution of values of an arithmetical function.

(Written jointly with R. Thangadurai) Indian J. Pure Appl. Math. 26 (10), 931-935, (1995).
11
On a question regarding visibility of lattice points.
(Written jointly with R. Balasubramanian) Mathematika 43, 155-158, (1996).
12
Addendum to A note on the distribution of values of an arithmetical function".
(Written jointly with R. Thangadurai) Indian J. Pure Appl. Math. 28 (1), 125-127, (1997)
13
A note on a question of Erd\"os.
- Exposition. Math.15, no. 4, 367-371(1997).
14 A note on sets having the Steinhaus property.
Note Mat. 16 (1996), No.1, 77-80 (1998)
15 On a question regarding visibility of lattice points-II. (Written jointly with Yong-Gao Chen) Acta Arithmetica 89 (1999) no. 3, (279-282)
16 Monochromatic solutions of Diophantine equations. (Written jointly with R. Thangadurai) Algebraic Number Theory and Diophantine Analysis (Graz., 1998) 1--9 de Gruyter, Berlin, 2000.
17 Further Remarks on Steinhaus sets. (Written jointly with R. Balasubramanian and R. Thangadurai) Publ. Math. Debrecen 57/ 3-4. 277-281 (2000)
18 Transcendental infinite sums. (Written jointly with N. Saradha, T.N.Shorey and R. Tijdeman) Indag. Math. (N.S.), 12 (1) 1--14 (2001).
19 On a theorem of Graham. - Expo. Math. 19, no. 4, 365 -- 367 (2001).
20 On the average of sum-of-odd-divisors function. (Written jointly with G. Coppola)
Current Trends in Number Theory
(proceedings of the international conf. held at HRI in November 2000),
1--15, Hindustan Book Agency, New Delhi, 2002.
21 On the average of the sum-of-the $p$- prime-divisors function. (Written jointly with G. Coppola and Anirban Mukhopadhyay) Acta Arith. 101, no. 4, 333 -- 338 (2002).
22 Corrigenda To Further remarks on Steinhaus sets'. (Written jointly with R. Balasubramanian and R. Thangadurai). Publ. Math. Debrecen, 60/ 1-2, 209-210 (2002).
23 On a question regarding visibility of lattice points-III. (Written jointly with Yong-Gao Chen) Discrete Mathematics, Vol. 259, 1-3, 251-256 (2003).
24 Remarks on the visibility problem in the function field case. (Written jointly with Francesco Pappalardi) C. R. Math. Acad. Sci. Soc. R. Can., 24 (3), 109--116 (2002).
25 Remarks on some zero-sum problems. (Written jointly with Purusottam Rath) Expo. Math., 21, no. 2, 185 -- 191 (2003).
26 Monochromatic configurations for finite colourings of the plane (Written jointly with Purusottam Rath) Note di Matematica, 22, no. 1, 59 --63 (2003/04).
27 On the sets of uniqueness of the distribution function of $\{ \xi (p/q)^n\}$} (Written jointly with P. Rath and N. Saradha) Acta Arithmetica, 119, no. 4, 307 -- 316 (2005).
28 Contributions to zero--sum problems (Written jointly with Y. G. Chen, J. B. Friedlander,S. V. Konyagin and F. Pappalardi) Discrete Mathematics, Vol. 306, 1, 1--10 (2006).
29 Davenport constant with weights and some related questions (Written jointly with P. Rath) Integers , 6, paper A 30, (2006).
30 Davenport constant with weights and some related questions - II (Written jointly with Y. G. Chen) Journal of Combinatorial Theory, Series A, 115, 178--184 (2008).
31 Some zero-sum constants with weights (Written jointly with R. Balasubramanian, F. Pappalardi and P. Rath) Proc. Indian Acad. Sci. (Math. Sci.), 118, 183 --188 (2008).
32 Generalizations of some zero-sum theorems (Written jointly with Chantal David and Jorge Jim\'enez Urroz) Integers , 8, paper A 52, (2008).
33 Weighted sums in finite Abelian groups (Written jointly with Mohan N. Chintamani, Bhavin K. Moriya and Prabal Paul) Uniform Distribution Theory, 3, no. 1, 105--110 (2008).
34 Remarks on some zero-sum theorems (Written jointly with Sanoli Gun and Purusottam Rath) Proc. Indian Acad. Sci. (Math. Sci.), 119, No. 3, 275--281 (2009).
35 Visibility in the plane (Written jointly with Andrew Granville) J. Number Theory , 129, Issue 10, 2335--2345 (2009).
36 Zero-sum problems with subgroup weights (Written jointly with A. A. Ambily and B. Sury) Proc. Indian Acad. Sci. (Math. Sci.), 120, No. 3, 259--266 (2010).
37 Number of weighted subsequence sums with weights in $\{1,-1\}$} (Written jointly with Mohan N. Chintamani) Integers , 11, paper A 36, (2011).
38 On weighted zero-sum sequences (Written jointly with David J. Grynkiewicz and Zhi-Wei Sun) Advances in Applied Mathematics, 48, 506--527, (2012).
39 Remarks on monochromatic configurations for finite colorings of the plane (Written jointly with Purusottam Rath) Note di Matematica, 32, no. 2, 83 --88 (2012).
40 Modifications of some methods in the study of zero-sum constants (Written jointly with Eshita Mazumdar) Integers , 14, paper A 25, (2014).
41 Erd\H{o}s-Ginzburg-Ziv theorem for finite commutative semigroups (Written jointly with Weidong Gao and Guoqing Wang) Semigroup Forum, 88, Issue 3, 555--568, (2014).
42 On monochromatic configurations for finite colorings (Written jointly with Yong-Gao Chen) Discrete Mathematics, Vol. 333, 106--109 (2014).
43 The polynomial method in the study of zero-sum theorems (Written jointly with Eshita Mazumdar) Int. J. Number Theory, Vol. 11, No. 5, 1451--1461 (2015).
44 On the $n$-color Rado number for the equation $x_1+x_2+\dots +x_k+c =x_{k+1}$ (Written jointly with L. Boza, S. Eliahou, J. Mar\'in, M. Revuelta and M. Sanz) Math. Comp., Vol. 85, 2011-2045 (2016).
45 Relation between two weighted zero-sum constants (Written jointly with Eshita Mazumdar and B. K. Moriya) Integers , 16, paper A 20, (2016).
46 On the finiteness of some $n$-color Rado numbers (Written jointly with L. Boza, S. Eliahou, J. Mar\'in, M. Revuelta and M. Sanz) Discrete Mathematics, Vol. 340, 39--45 (2017).
47 The analog of the Erd\H{o}s distance problem in finite fields (Written jointly with Anirban Mukhopadhyay and M. Ram Murty) Int. J. Number Theory , To appear.

### Expository Articles:

1. Colours in the fields coming in handy: a walk in the garden of Ramsey-type theorems. Bona Mathematica, Vol. 6, No 4, 7-- 14 (1995).
2. Lattice points in spheres. Bulletin of the Allahabad Mathematical Society, Vol. 8-9, (1993-94), 1-- 13 (1997).
3. The average behaviour of the number of solutions of a Diophantine equation and an averaging technique.  Number Theory: Diophantine, Computational and Algebraic Aspects, Proceedings of the International Conference held at Eger, Hungary (Eds. K Gy\"ory, A. Peth\"o and V. T. S\'os), Walter de Gruyter GambH \& Co., Berlin, New York, 1--8 (1998).
4. On some questions related to integer lattice points on the plane. Number Theory and Applications (Japanese) (Kyoto 1997) S\=urikaisekikenky\=usho K\=oky\=uroku No. 1060, (1998), 231-237
5. A Problem of Steinhaus concerning the existence of a plane set with a certain property. Number Theory and its Applications (Kyoto, 1997) 1-5 Dev. Math.,2, Kluwer Acad. Publ., 1999.
6. The Early Reciprocity Laws: From Gauss to Eisenstein. Cyclotomic Fields and Related Topics. edited by S. D. Adhikari, S. A. Katre and Dinesh Thakur, Published by Bhaskaracharya Pratishthana, Pune, 55-74 (2000).
7. Partition function congruences: Some flowers and seeds from Ramanujan's garden. (Written jointly with Anirban Mukhopadhyay). Expo. Math. Vol. 19, 193-- 211 (2001).
8. Transcendental infinite sums and some related questions. Number Theory and Discrete Mathematics (proceedings of international conf. on number theory held at Chandigarh in October 2000), 169-- 178, Hindustan Book Agency, New Delhi, 2002.
9. Arithmetic nature of sums of certain convergent series. (Jointly with N. Saradha). Resonance , Vol. 7, No. 11, 35 -- 46, 2002.
10. Siegel's Theorem: Finiteness of integral points. (Jointly with D. S. Ramana). Elliptic Curves, Modular Forms and Cryptography, Proceedings of the advanced instructional workshop on Algebraic Number Theory, HRI, Allahabad, 2000 (Eds. A. K. Bhandari, D. S. Nagaraj, B. Ramakrishnan, T. N. Venkataramana), Hindustan Book Agency, New Delhi 2003.
11. Some questions regarding visibility of integer lattice points in $I\!R^d$}. Proceedings of the Session in analytic number theory and Diophantine equations (Bonn, January -- June 2002) (Eds. D. R. Heath-Brown and B. Z. Moroz), Bonner Mathematische Schriften, {\bf Nr. 360}.
12. Zero-sum problems in combinatorial number theory. (Jointly with P. Rath). The Riemann Zeta Function and Related Themes: Papers in Honour of Professor K. Ramachandra, Proceedings of international conf. held at National Institute of Advanced Studies, Bangalore, 13-15 December 2003 (Eds. R. Balasubramanian and K. Srinivas), Ramanujan Mathematical Society Lecture Notes Series, Number 2, 2006.
13. Some combinatorial group invariants and their generalizations with weights. (jointly with R. Balasubramanian and P. Rath). Additive Combinatorics (Eds. Granville, Nathanson, Solymosi) 327 --335, CRM Proceedings and Lecture Notes, Volume 43, American Mathematical Society, 2007.
14. An extremal problem in lattice point combinatorics. (jointly with Stephan Baier and Purusottam Rath). Diophantine Equations, (N. Saradha Ed.) 19 --32, Tata Institute of Fundamental Research, Publisher: Narosa, New Delhi, 2008 .
15. A problem on the fractional parts of the powers of $3/2$ and related questions. (jointly with Purusottam Rath). Number Theory and Discrete Geometry, Proceedings of the International Conference on Number Theory \& Discrete Geometry held in honour of Professor R. P. Bambah, held at Department of Mathematics, Panjab University (Chandigarh, November 30-December 3, 2005) (Eds. R. Balasubramanian, S. G. Dani, P. M. Gruber and R. J. Hans-Gill), 1--12, Ramanujan Mathematical Society Lecture Notes Series, Number 6, 2008.
16. A pearl of Number Theory, the theorem of van der Waerden. Mathematics Newsletter, Published by Ramanujan Mathematical Society, Vol. 18, No 2, 58--62, 2008.
17. The relation between two combinatorial group invariants : History and ramifications. (Text of the Nineteenth Hansaraj Gupta Memorial Award Lecture at the IMS Conference 2008). Math. Student, Vol. 77, 131-- 143 (2008).
18. Some omega results and related questions in the research work of Pillai. (An article about S. S. Pillai's work). Collected works of S. Sivasankaranarayana Pillai (Eds. R. Balasubramanian and R. Thangadurai), l--lvi, Published by Ramanujan Mathematical Society, India, 2010.
19. The Cauchy-Davenport theorem: various proofs and some early generalizations. (jointly with M. N. Chintamani, Geeta and B. K. Moriya). Math. Student, Vol. 79, 109-- 116 (2010).
20. Simplices with monochromatic vertices: Reflections on some problems in Euclidean Ramsey Theory. (jointly with Purusottam Rath). Number Theory, Proceedings in honour of Professor T. C. Vasudevan on his sixtieth birthday, (Eds. M. Manickam and B. Ramakrishnan), 1--5, Ramanujan Mathematical Society Lecture Notes Series, Number 15, 2011.
21. Erd\H{o}s, Ginzburg and Ziv Theorem: various proofs and some generalizations. (jointly with Eshita Mazumdar and Sneh Bala Sinha). Proceedings of the 21st Annual Conference of the Jammu Mathematical Society, (Eds. K. S. Charak, R. Kumar), 41--48, University of Jammu.
22. Schur's theorem in combinatorial number theory: Some generalizations. J. Indian Math. Soc. (N.S.) 2013, Special Volume to commemorate the 125th birth anniversary of Srinivasa Ramanujan, 1--10.
23. An extremal problem in combinatorial number theory. Proceedings of an International Conference in Celebration of the 125th Anniversary of Ramanujan's Birth; University of Delhi, 17-22 December 2012, (Eds. Bruce C. Berndt and Dipendra Prasad), 1--11, Ramanujan Mathematical Society Lecture Notes Series, Number 20, 2013.
24. Some omega results and related questions connected to R. Balasubramanian's mathematical work. (Jointly with D. S. Ramana). Highly Composite: Papers in Number Theory, (Eds. V. Kumar Murty and R. Thangadurai), 1--11, Ramanujan Mathematical Society Lecture Notes Series, Number 23, 2016.

### Monographs / Books

1. An introduction to Commutative Algebra and Number Theory. Narosa Publishing House, (1999).
2. Aspects of Combinatorics and Combinatorial Number Theory. Narosa Publishing House, (2002).

### Volumes edited:

1. Current Trends in Mathematics and Physics. A tribute to Harish-Chandra. Narosa Publishing House, New Delhi, 1995.
2. Cyclotomic Fields and Related Topics. (Edited jointly with S. A. Katre and Dinesh Thakur), Published by Bhaskaracharya Pratishthana, Pune, 2000.
3. Current Trends in Number Theory (proceedings of the international conf. held at HRI in November 2000).  (Edited jointly with S. A. Katre and B. Ramakrishnan). Hindustan Book Agency, New Delhi, 2002.
4. Number Theory (proceedings of the international conf. held at The Institute of Mathematical Sciences, Chennai, in January 2002). (Edited jointly with R. Balasubramanian and K. Srinivas). Ramanujan Mathematical Society Lecture Notes Series, Number 1, 2005.
5. Number Theory and Applications (proceedings of the international conferences on Number Theory and Cryptography). (Edited jointly with B. Ramakrishnan). Hindustan Book Agency, New Delhi, 2009.

### Ph. D. Works supervised

3. Dr. Purusottam Rath
4. Dr. Anupama Panigrahi
5. Dr. Bhavin. K. Moriya
6. Dr. Mohan Chintamani
7. Dr. Eshita Mazumdar

### Other experiences and recognitions

1. A fellow of the National Academy of Sciences, India.
2. Delivered the Nineteenth Hansraj Gupta Memorial Award Lecture' of Indian Mathematical Society in 2008.
3. Received the M K Singhal Memorial Award for the year 2013-14 by the Indian Science Congress Association.
4. Received Distinguished Faculty Award of Homi Bhabha National Institute (HBNI) in 2015.
5. Received D. Sc. (Honoris Causa) from Kazi Nazrul University, Asansol, West Bengal, India, in 2016.
6. A member of National Board of Higher Mathematics (NBHM), India.
7. A member of the editorial board of The Journal of the Indian Mathematical Society'.
8. A member of the editorial board of Proc. Indian Acad. Sci. (Math. Sci.)'