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I already asked a similar question http://mathoverflow.net/questions/48742/is-there-natural-integration-constant-closed, but it seems it was not understood properly, so I am trying now to formulate it differently.

Is there an operator R[f] that plays in integral world the same role as Ramanujan sum plays in the world of series?

It is known that Ramanujan sum plays the role of natural integration constant for discrete integration, that is if F(x) is the discrete integral of f(x), usually postulated that

$$F(0)=\sum^{\Re}f(x)$$

This yields the functions which have some very useful properties, for example, Bernoulli polynomials (which are the results of discrete integration of power function), the Hurwitz Zeta function (which is generalization of Bernoulli polynomials), etc. Discrete integrals which normalized with Ramanujan sum (or equal method) called "balanced" (as opposed to some functions normalized without it, for example to be zero in zero such as Harmonic numbers).

One of the properties of balanced functions is that $$\int_0^1 F(x)\ dx=0$$

To outline the properties of such operator in integral world we should admit that such operator should be symmetric against zero (unlike the Ramanujan sum).

It is also highly desirable (if possible) that R[exp(x)]=1 thus making integral of exponent itself exponent and thus invariant against differintegral operator. This would allow to provide integration constants for trigonometric functions that would work in agreement with known expressions for differintegral:

$$D^q \sin x=\sin \left(x+\frac{q\pi}{2}\right)$$

P.S. See my own answer below.

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# Is there integral analog of Ramanujan sum?

I already asked a similar question http://mathoverflow.net/questions/48742/is-there-natural-integration-constant-closed, but it seems it was not understood properly, so I am trying now to formulate it differently.

Is there an operator R[f] that plays in integral world the same role as Ramanujan sum plays in the world of series?

It is known that Ramanujan sum plays the role of natural integration constant for discrete integration, that is if F(x) is the discrete integral of f(x), usually postulated that

$$F(0)=\sum^{\Re}f(x)$$

This yields the functions which have some very useful properties, for example, Bernoulli polynomials (which are the results of discrete integration of power function), the Hurwitz Zeta function (which is generalization of Bernoulli polynomials), etc. Discrete integrals which normalized with Ramanujan sum (or equal method) called "balanced" (as opposed to some functions normalized without it, for example to be zero in zero such as Harmonic numbers).

One of the properties of balanced functions is that $$\int_0^1 F(x)\ dx=0$$

To outline the properties of such operator in integral world we should admit that such operator should be symmetric against zero (unlike the Ramanujan sum).

It is also highly desirable (if possible) that R[exp(x)]=1 thus making integral of exponent itself exponent and thus invariant against differintegral operator. This would allow to provide integration constants for trigonometric functions that would work in agreement with known expressions for differintegral:

$$D^q \sin x=\sin \left(x+\frac{q\pi}{2}\right)$$