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Vivek V. Rane

Publications and source records attributed to Vivek V. Rane.

12 recordsLinked to original sources
arXiv2013

Precise Definition And Analyticity of r-th order partial derivative of zeta(s,alpha)

S.Ramanujan[1] was aware of power series expression in alpha of zeta(s,1-alpha)for complex s and for 0<=alpha<1,which he did not explore very far.Author[2] had derived power series expression in alpha of zeta(s,1+alpha)for complex α with |alpha|<1 and had shown in author[3] that the power series of zeta(-n,alpha) for integral n>=0,is a polynomial in α .On this backdrop, we give here the precise definition of r-th order partial derivative w.r.t s of zeta(s,alpha) in terms of power series in alpha for complex s and alpha . We also discuss the analyticity of r-th order partial derivative w.r.t s of zeta(s,alpha) as functions of s and alpha and show that partial derivatives w.r.t s and alpha commute.We compute partial derivative w.r.t alpha of r-th order partial derivative w.r.t. s of zeta(s,alpha)for s=0 and s=1.We discuss the power series in s of zeta(s+1,alpha) for complex alpha.

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arXiv2012

Classical Theory of Fourier Series:Demystified and Generalised

For a Riemann integrable function on an interval and for a point therein,we define 'Fourier series at the point on the interval' and bring out how and when the function element becomes expressible as Fourier series.In this process,we also generalise the theory by bringing in such concepts as finite Fourier series,right/left hand Fourier series.We also sum up subseries corresponding to terms in an arithmetic progression,of the basic Fourier series.

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arXiv2011

Functional Equations of Tornheim Zeta Function and the alpha-calculus of r-th order derivative of Zeta(s,alpha)

For the Tornheim double zeta function T(s1,s2,s3) of complex variables,we obtain its functional equations,which are new.Using the calculus of r-th order derivative of zeta(s,alpha) as a function of alpha(developed in author[7])as the tool,we obtain with ease an expression for T(s,s,s) for any complex s;and evaluate T(n,n,n) for any integer n>=1;and also evaluate T(n1,0,n2)for integers n1,n2>1 with n1+n2 odd.We obtain for Tornheim zeta function,the counterparts of Euler's formula for zeta(2n)and its analogue for zeta(2n+1),where Zeta(s) is Riemann zeta function.

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arXiv2011

Fourier Series Of the Derivatives of Hurwitz and Lerch Zeta Functions

As a function of second variable, we identify the Fourier series of Hurwitz zeta function and its derivatives on the unit interval. Consequently, we obtain results based on the formula for Fourier coefficients and also on Parseval's theorem. We do likewise in the case of Lerch's zeta function and its derivatives.

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arXiv2010

Gamma,Psi,Bernoulli Functions via Hurwitz Zeta Function

Using three basic facts concerning Hurwitz zeta function,we give new natural proofs of the known results on Bernoulli polynomials,gamma function and also obtain Gauss' expression for Psi function at a rational point,all in a unified fashion.We also give a new proof of the relation between log gamma and derivative of the Hurwitz zeta function,including that of Stirling's expression for log gamma.

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arXiv2009

Instant Multiple Zeta Values at Non-Positive Integers and Bernoulli Functions

We give an instant evaluation of multiple Zeta function at non-positive integers by elementary methods and discuss the Fourier theory (on unit interval) of the product of Bernoulli polynomials.We also show that the polynomial expression for Hurwitz Zeta function(at non-positive integral values of the first variable) and the polynomial expression for Bernoulli polynomials are equivalent.

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arXiv2008

Demystification of Taylor,Laurent coefficients of Lerch,Hurwitz Zeta functions and Dirichlet L-Function at Unity and Zero and their Bounds

Using elementary methods,we obtain simple,explicit expressions and bounds of higher order derivatives of Hurwitz zeta function and consequently those of Dirichlet L-function and also,of Lerch's Zeta function at unity (and at Zero too)and also obtain their interrelations.We also state elementary complete forms of approximate functional equations of Hurwitz zeta function and Dirichlet L-function in the critical strip.

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arXiv2008

Divisor Problem and an Analogue of Euler's Summation Formula

By simple elementary method,we obtain with ease,a highly simple expression for the remainder term of the divisor problem and use it to obtain an Euler-Maclaurin analogue of summation involving divisor function.We also obtain a relation connecting the remainder term of the divisor problem and the remainders of approximate functional equations for Riemann zeta function and its square,for positive values of arguments.

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arXiv2008

Instant Evaluation and Demystification of zeta(n),L(n,chi) that Euler,Ramanujan Missed III

We show that for a non-positive value of the first variable,Hurwitz zeta function becomes a polynomial in the second variable. We show this, using 'integration approach', instead of 'power series approach', which we had resorted to, in our earlier paper with the same title. This, in particular, explains why Riemann zeta function at positive even integer arguments, can be evaluated and why it cannot be evaluated explicitly at positive odd integer arguments.

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arXiv2008

Instant Evaluation and Demystification of zeta(n),L(n,chi) that Euler,Ramanujan Missed - II

For Hurwitz zeta function, we obtain power series expression in second variable for its higher order derivatives (with respect to first variable) at non-positive integer arguments and consequently obtain rapidly decreasing series expression for Riemann zeta fuction at positive odd integer arguments. Further, we obtain corresponding results for Dirichlet L-series. We also a unified proof of various classical identities involving Riemann zeta values.

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arXiv2008

Instant Evaluation and Demystification of $ζ(n),L(n,χ)$ that Euler,Ramanujan Missed - I

For Hurwitz Zeta function,we consider its Taylor series expansion about various points as an analytic function of second variable in appropriate discs.We show that these Taylor are all polynomials in second variable for a non positive integral argument in first variable.On using functionalequations this results in instant evaluation of Riemann Zeta function at positive even integral values of its argument and of Dirichlet L series at positive integral values of its argument,when the argument and the corresponding Dirichlet character are both even or both odd.We also obtain finite sum expression for any Dirichlet L series,when its argument is one.We also deal with Lerch's Zeta function on similar lines.

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