At which point is the magnetic field in a current carrying loop the strongest? At which point inside a circular current carrying loop the strongest? Some sources say it is at the centre and some say it is at the circumference.
And how can we mathematically prove this?
And also, is the magnetic field line through the centre of the loop exactly a straight line? Because this diagram shows so:

However if the field lines are concentric circles centered at the circumference then they can never become a straight line.
 A: As a rough estimate for the behaviour, I have plotted a graph.

Taking a slice of the loop, the field from the left and right current elements fall off like 1/r^2, here I have modelled the graph such that r is radius of the loop, and x is the displacement from the origin.
See how inside the loop, the magnetic field is actually weakest in the center, and approaches infinity after we reach the radius of the loop
Outside the loop, the magnetic field falls off rapidly, which is the expected behaviour.
The behaviour of this graph is for infinitely thin currents, however I believe the generalisation to volume currents is similar, with this function having a domain restriction once we reach the wires radius to avoid infinite magnetic fields.
I could be wrong, but I am fairly confident that this is a good approximation (obviously ignoring   there is a full circle, not 2 current elements). Focusing on the  rough shape, and not the actual numbers atleast.
A: The magnetic flux on an electric current wire loop is the same case of that of a straight wire magnetic concentric rings around the wire. The only difference is that if we make a loop with the straight wire the concentric rings will become longitudinally tangent with each other towards the center of the loop and stretch out and become more linear towards the loop center. Also the more the center we approach the more linear the flux lines become and more homogeneous the field is i.e. same strength B (i.e. density of field lines) magnetic field, thus same number of lines passing through per unit area and the more straight they become thus zero curl $∇ × B=0$ for the flux at the center of the loop.
The field closer to the rings' wire surface is stronger than that on the center and not homogeneous.
By using Bio-Savart Law the field strength exactly at the center of the current loop of radius R can be calculated as:
$$
B=\frac{\mu_{0} I}{2 R}(\text { at center of loop })
$$

Where the field at the center is more homogeneous but this equation above is in practice an approximation for distances $r<R$ since the field strength near the conductor wire becomes inhomogeneous there. Nevertheless, it can be used by replacing in the above equation  the radius R value with the given r value to find the corresponding dB value (see illustration above).
Much more homogeneous magnetic field and on a larger area of the loop can be achieved by cascading many loops in series and forming a helix spiral thus an electric solenoid:

A: To apply the Biot-Sevart formula to a circular loop of current, I put the origin of an xy system at the center of the loop of radius, R.  I wanted the B field at a point on the x axis (in the plane of the loop) at x = a. A line segment in the first quarter of the loop (for a CCW current) would be:  (dL) = R(dθ)(- sin(θ)i + cos(θ)j).  The vector to the field point would be: r = - (R cos(θ) – a)i  - r sin(θ)j.  The contribution to the field would be (1E-7)I(dL) x r /$r^3$ (with the x denoting a vector product). To get numerical results, I set R = 1 m, and  I = 1 amp, with different values of , a.  I have a spreadsheet with a macro which does numeric integration, and found that the field was a minimum at the center of the loop at 6.283E-7 T (in the z direction), and increased going out along the x axis, getting large as it approached the wire.
