I am interested in finding the variation of argument of a certain analytic function $g(z)$ on a region of the form $\frac{1}{2}\leq \sigma \leq \sigma_1$, $t_0\leq t \leq T$ where $\sigma$ stands as usual for the real part, and $t$ for the imaginary part.
However, my function has zeros on the line $\sigma=\frac{1}{2}$, $0\leq t \leq T$, and thus I cannot apply the argument theorem. I asked my professor, and he told me that there was a lemma by Hardy and Littlewood if I am not mistaken, that gives a result of the form:
$$2\pi \sum_{\alpha<\sigma_0}(\sigma_0-\alpha)=\int_{t_0}^Tlog|g(\sigma_0+ti)|dt-\int_{\sigma_1}^{\sigma_0}arg(g(\sigma+Ti))d\sigma-\int_{t_0}^Tlog|g(\sigma_1+ti)|dt+\int_{\sigma_1}^{\sigma_0}arg(g(\sigma+t_0i))d\sigma.$$
Where $\sigma_0<\frac{1}{2}$, and the sum is over the $\alpha$, the real parts of the roots of $g(z)$.
However I have not been able to find such a lemma. Does anybody have a reference where I can consult this, or the name of this lemma?
The result you are referring to is likely the `Littlewood Argument Principle', also known as the 'Littlewood Lemma'. (I prefer the first name: its abbreviation, LAP, could also be taken as an abbreviation for "Logarithmic Argument Principle" or "Lap Around Perimeter".) It has its origin in:
J. E. Littlewood, On the zeros of the Riemann zeta-function, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 22, issue 3, 20 September (1924), pp. 295--318.
It also appears in various forms in:
E. C. Titchmarsh, The Theory of Functions, Oxford University Press, 2nd. edition. [See §3.8]
E. C. Titchmarsh, The Theory of the Riemann Zeta-function, Oxford University Press, 1986, part of the Oxford science publications (Revised second edition by D.R. Heath-Brown, 2007). [See §9.9]
H. Iwaniec, Lectures on the Riemann Zeta Function, American Mathematical Society, 2014, vol. 62 of the University Lecture Series. [See §21]