Here is a simple and commonly used Gautam Swami mantra / shloka with meaning:

1) Short mantra

Sanskrit: ॐ गौतमस्वामिने नमः।

Transliteration: Om Gautam Swāmine Namah।

Meaning (arth):

• I bow to Gautam Swami.
• May I receive right faith, right knowledge, and humility like him.

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2) Slightly longer shloka

Sanskrit: गौतमगणधरं वन्दे, ज्ञानदर्शनदायकम्। सर्वसंकल्पसंक्लेशं, नाशयित्वा सुखावहम्॥

Transliteration: Gautama–gaṇadharaṃ vande, jñāna–darśana–dāyakam। Sarva–saṅkalpa–saṅkleśaṃ, nāśayitvā sukhāvaham॥

Meaning (arth) – simple words:

• I bow to Gautam Ganadhar (chief disciple of Mahavira),

who gives right knowledge and right vision,

• who removes all doubts, worries and confusions,

and brings true happiness and peace.

This mantra is accepted and used with devotion in both Śvetāmbara and Digambar traditions; the wording may vary slightly in books or stavans, but the bhāva (feeling) is the same – deep respect to Gautam Swami as an ideal disciple of Bhagwan Mahavir.

You can read more about Gautam Swami here