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In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from which follows a formula that connects the derivatives of the Riemann zeta - function at the negative integers to the Ramanujan summation of the divergent Euler sums $\sum_{n\ge 1} n^kH_n, k \ge 0$, where $H_n= H_n^{(1)}$. Further, we express our results on the Ramanujan summation in terms of the classical summation process called the Borel sum., Comment: 25 pages

We study three special Dirichlet series, two of them alternating, related to the Riemann zeta function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at poles). These values are given in terms of Bernoulli and Euler numbers., Comment: 15 pages, To appear in the Hardy-Ramanujan journal

We study a special Dirichlet series studied before by Apostol and Matsuoka and specify its values at negative integers. These values are related to a certain convolution of Bernoulli numbers

We connect and generalize Matiyasevich's identity #0102 with Bernoulli numbers and an identity of Candelpergher, Coppo and Delabaere on Ramanujan summation of the divergent series of the infinite sum of the harmonic numbers. The formulae are analytic continuation of Euler sums and lead to new recursion relations for derivatives of Bernoulli numbers. The techniques used are contour integration, generating functions and divergent series., Comment: 10 pages

It is well known that the repeated square and multiply algorithm is an efficient way of modular exponentiation. The obvious question to ask is if this algorithm has an inverse which would calculate the discrete logarithm efficiently. The technical hitch is in fixing the right sign of the square root and this is the heart of the discrete logarithm problem over finite fields of characteristic not equal to 2. In this paper a couple of probabilistic algorithms to compute the discrete logarithm over finite fields are given by bypassing this difficulty. One of the algorithms was inspired by the famous 3x+1 problem., Comment: 15 pages

Currently the circle and the sieve methods are the key tools in analytic number theory. In this paper the unifying theme of the two methods is shown to be the Ramanujan - Fourier series., Comment: 18 pages, Based on a talk given at the Conference in Number Theory held at NIAS, Bangalore, India during 13-15, December 2003 in honor of Professor K. Ramachandra

It is noted that an efficient algorithm for calculating a $p$-adic height could have cryptanalytic applications., Comment: 4 pages

A prime p is called Sophie Germain prime if 2p+1 is also prime. A formula for the density of such primes is given in a more general setting using a new approach. This method uses the Ramanujan-Fourier series for a modified von Mangoldt function. The proof remains heuristic as interchange of certain limits has not been justified. Experimental evidence using computer calculations is provided for the plausibility of the result., Comment: 11 pages

In this brief note we bring out the analogy between the arithmetic of elliptic curves and the Riemann zeta-function., Comment: 7 pages

In this note we point out the fact that the proper conceptual setting of quantum computation is the theory of Linear Time Invariant systems. To convince readers of the utility of the approach, we introduce a new model of computation based on the orthogonal group. This makes the link to traditional electronics engineering clear. We conjecture that the speed up achieved in quantum computation is at the cost of increased circuit complexity., Comment: 8 pages

In this note we point out links between the Shimura - Taniyama conjecture and certain ideas in physics. Since all the seminal references are by strange coincidence Japanese we wish to call this the Japanese approach. The note elaborates on some inspired comments made by Barry Mazur in his popular article "Number theory as gadfly.", Comment: Latex2e, 4 pages

In this paper the ideas of Rota and Ramanujan are shown to be central to understanding problems in additive number theory. The circle and sieve methods are two different facets of the same theme of interplay between probability and Fourier series used to great advantage by Wiener in engineering., Comment: Latex2e, 4 pages

The entropy of an orthogonal matrix is defined. It provides a new interpretation of Hadamard matrices as those that saturate the bound for entropy.It appears to be a useful Morse function on the group manifold. It has sharp maxima and other saddle points. The matrices corresponding to the maxima for 3 and 5 dimensions are presented. They are integer matrices (upto a rescaling)., Comment: 7 pages, minor typographical errors corrected and conclusion added

Bohr and Rosenfeld carried out an analysis of the consequences of field theory commutation relations. In this note the analysis is sharpened. A conjecture of Heisenberg that volume is quantized is shown to be a consequence of the second quantization of gauge fields. A way to generalize the equations of physics to include the Planck length is indicated., Comment: 7 pages

Non-Abelian Gauss law is interpreted in terms of area bits described in a local frame which fit together into closed surfaces and the Non-Abelian Stokes law in terms of length bits described in a local frame which fit together into closed loops. A new equation relating the area variables and the phase space variables (or equivalently, angular momentum variables of the lattice Yang-Mills theory and phase space variables of the continuum theory) is obtained. Canonical quantization applied to these variables implies area quantization. A complete orthonormal basis of states satisfying the Gauss constraint is obtained.It has the interpretation of quantized area bits with undefined orientations and edges but fitting together into closed surfaces., Comment: 8 pages, revtex

The analogy between General Relativity and monopole physics is pointed out and the presence of a 3-cocycle which corresponds to a source leads to discretization of field momentum. This is analogous to the same phenomena in monopole physics., Comment: 3 pages